Both Herman Yandell and Elmer Snell from the second episode of MS IGLOO 2: The Gravity Front claim to have very temperamental machines (a tank and a Zaku, respectively) that break down despite regular maintainance saying they should still run. This seems to imply that the machines are empathic to their pilots, since it serves to tell them that the enemy they're looking for is not in the battlefield they were going to, and they want to save their strength for their fated encounter.
Spike of Cowboy Bebop averts this, much to the disgust of the mechanic that gave him the Swordfish. He actually first reveals his insensitivity when he destroys a malfunctioning Betamax player by kicking it.
Spike: My ship works when I kick it...
Forge from the X-Men sometimes exhibits this ability, depending on the writer.
Characters in Sin City often determine the state of the cars they're in, including the engine types, just by starting them.
In the live-action Speed Racer movie, Rex teaches Speed to "drive, not steer" by listening to the feel of the car. Speed later uses this ability to intuitively jump-start the Mach 6 in the middle of the final Grand Prix.
Subverted in The Empire Strikes Back; the Millenium Falcon's hyperdrive repeatedly fails despite Han Solo's insistence that it's fixed.
In Star Trek Insurrection, Captain Picard detects that the ship's torque sensors are slightly out of alignment because "they don't sound right."
Justified when we find out in the next movie that Picard once suffered from an illness that cranks up the sufferer's sensitivity to the point that every minor sound was agony. He got it cured, but it stands to reason that it left him with extremely acute hearing (by human standards).
Jake Holman from The Sand Pebbles feels more comfortable around machines than around people. Given the way he affectionately speaks to the ship's steam engine, it almost qualifies as Cargo Ship.
Engines still provide thrust that you'll be feeling in the cockpit with the damping dialed down.
Wedge Antilles is very, very good at this, as well as possessing Improbable Piloting Skills. These factors together make some people in-universe question the Normal part of his Badass Normal status. He can hear if his X-wing's engines have drifted out of alignment. (This may also have something to do with the Incom T-65's standard engines, which are mentioned in several EU sources as having a unique and almost musical sound — this being the case, perhaps what Wedge is noticing is that the engines have literally gone out of tune.)
In one of the EU novels the Millennium Falcon gets a complete overhaul by New Republic techs (estimating that they replaced nearly 20% of the ship). Han spends the next several hours going through the ship and loosening bolts, stripping wires and generally messing things up, because "Those rattling noises are how he knows he's pushing her too hard".
In Sandy Mitchell's Warhammer 40000Ciaphas Cain novel Death of Glory, a character born in the ship is able to tell something goes wrong from Cain's quarters, long before the alarm rings. Cain compares it to how himself instinctively "feels" an underground environment.
Live Action TV
As the page quote shows, Scotty from the original Star Trek regularly invoked this trope.
As did Kirk, once. He claimed to recognize every noise the ship could make, even if damaged.
When Scotty appeared in Star Trek: The Next Generation, he pointed out that he didn't like the Enterprise D because it doesn't cause the vibrations that his Enterprise had.
Picard did this once, too, able to tell by the sound when (insert TNG techspeak here) was off by three microns (that's really small.) However, it was revealed later that he'd had a genetic illness in which his senses were heightened to the point that every small sound was agony to him, presumably dealt with by Trekkian super-science later. You have to wonder if it's an intentionally-invoked Harsher in Hindsight moment - was it a captain's ship senses tingling, or do we know now that all along he's had this painful affliction that's less "gone away" and more "gone from agonizing down to just bearable?"
William Adama has a very personal relationship with the Battlestar Galactica, which goes beyond the relationship a captain has with his vessel. This is especially noticeable in the last episodes of season 4, when Adama refuses to use Cylon tech to repair the ship, not only because of the security risks involved, but also because it would turn the ship into something not entirely his — and not entirely hers; his comment on the matter is that "[s]he won't know what she is anymore."
Boomer's comments about the mechanics of a Cylon raider are obvious (to the audience) allusions to her own Cylon nature, but definitely fall along the vein of this trope. (Perhaps more naturally so than most, in fact; Boomer might well have Machine Empathy with the Raider because she's from the same 'family' of machines. Chief Tyrol, too.)
In Firefly, Kaylee has a Machine Empathy relationship with Serenity; she can instinctively know just by looking, hearing or feeling what the ship is doing and what's wrong with it. The Captain Mal and the Ace Pilot Wash have expressed similar sentiments, as has The Empath River. Not for nothing is Serenityconsidered part of the crew.
Mal: Know what the first rule of flyin' is? [...] Love. You can know all the math in the 'Verse, but you take a boat in the air that you don't love, she'll shake you off just as sure as a turn in the worlds. Love keeps her in the air when she oughta fall down, tells ya she's hurtin' 'fore she keens. Makes her a home.
In the Farscape episode "Back and Back and Back to the Future", the following exchange occurs between Zhaan and Rygel. Though Moya is a living being, the fact that she's a ship means Zhaan would have to have a certain level of Machine Empathy to detect the problem.
Zhaan: Rygel. You've been aboard Moya longer than anyone else except Pilot. You know her sounds and her rhythms. Just stop and listen to her for a moment.
Rygel: Moya sounds fine.
Zhaan: Does she? Not to me. Something feels... out of balance.
Dean from Supernatural seems to have this in regards to his Impala.
Not surprise, it's been the boy's literal home for many a year. In fact, this concept becomes very important as the show goes on.
Doctor Who The Doctor fells this towards his TARDIS (a.k.a. "You Sexy Thing").
The trope namer is the Role-Playing GameParanoia, where "Machine Empathy" is the name of a mutation that allows a character to influence how robots and The Computer react to him (though this ability falls under the Technopath trope). Since this could give a person undue influence with the all-powerful Computer, it's also the one mutation that, if found out, will get you and your clone family terminated and your gene template erased with extreme prejudice.
Presumably the theory behind the rule in one game that you get a bonus on vehicle operation rolls if you've been operating the same vehicle for at least two years. Not the same make and model, the exact same vehicle.
Implied with the Mechanic non-player character in the Shadowrun second edition rulebook. She is described as someone who prefers the "company" of machines over humans because they don't talk back and she understands them. (Perhaps Truth in Television; in Real Life, many who incline to the technical and mechanical have been known to say the same.)
The "Loved" trait in the Serenity RPG - the crew of a ship has cared for and operated it for so long that they know its processes on a nigh-quantum level. This lets them spend their own Plot Chips on behalf of the ship.
In Knights Of The Old Republic II, the Jedi Exile discovers they have this talent. Not only can they sense the presence of droids and ships through it's unique energy patterns, this allows them to immediately detect and diagnose any mechanical problem it's suffering.
Joker exhibits this in Mass Effect towards both incarnations of the Normandy. In the second game, he frequently gets into heated arguments with EDI over all the changes he impliments whilst flying, which run contrary to the design specifications, simply because that's how "his baby" works best. Keep in mind, since EDI is the artificial intelligence operating the ship, he's effectively telling her he knows her "body" better than she does!
Possibly justified. EDI only knows what her sensors tell her, and her definition of "best" may vary from Joker's.
In thisFreefall strip, Florence mentions the phenomena as she listens to learn how the Savage Chicken sounds in flight.
Chakona Space gives us Chakat Goldfur, one of the Star Corps' best technicians. Hir Talent even gets lampshaded at different points.
The Venture Brothers: Brock Samson has demonstrated the ability to sense when someone's in his car even if he's not physically present. Hank, upon witnessing a demonstration of this talent (which, despite the presence of superpowers and magic in this setting, seems to be purely mundane), comments to another character "I've seen him do that from a continent away."
Truth in Television: it's possible to invoke this with a car or whatever that you're very familiar with, especially if it's a problem it had before.
After driving a car long enough you can simply feel how much to turn, how much brake you need to apply, etc. You become one with the car. (You also know, for example, when the electrical system stops working in a rainstorm, just how to tap the fusebox with your boot so the headlights, wipers, and instrument cluster all start working again.)
Anyone who is Driving Stick will end up with muscle memory responses to audio cues (when the engine sounds like that it's time to shift up/down, no thinking required), and vibrations picked up through the feet.
The same goes for PCs and just about any other reasonably complicated machine that can be customized by its owner (or simply needs repairing a lot).
Also quite common for lab workers- at first all the lab equipment is just a noisy, humming, hissing, and beeping mess, but after a while you learn where each sound comes from and what to do when a sound stops or changes. Some critical equipment is even built to chirp every few seconds so that if it breaks you'll know right away- you learn to ignore the sound when it's working but it's very noticeable when the alert stops.
Manufacturing plants, too. You stand in the middle of the plant trying to place which "clunk" or "clank" sounds different or is not occurring. More experienced employees catch each other's eye and tilt their heads back and forth like dogs, while newbies watch, confused.
And sysadmins. Eventually, at least if you're fit for the job, you get good enough at it to notice how the background noise subtly changes texture when one fan out of sixty stops running. If you want to know which people at the party spend their days in a datacenter, bring something that makes a loud, insistent beep, and watch to see who squints and looks around every time you set it off.
Lifer light-aircraft pilots almost invariably develop this to the point of being contagious, even to people with no demonstrable mechanical knowledge.
Any mechanic worth their tool box will develop this with enough experience on a given model.
The US Air Force has a special division nowadays with specific machines to listen to other machines, so this happens even without a human operator.
This is extremely noticeable with firearms. This is the whole reason soldiers and even some police train to death with one or two weapons, so they have this ability. Considering the raw amount of energy going through a firearm, its no wonder you're going to get some sort of tactile feedback.
Very important with guns. If the kick feels unusually weak or the bang sounds wrong, you really want to inspect the weapon before you pull the trigger again, in case there's a dud round sitting in the barrel or some other malfunction.