So, the ad Todd and Riley found was meant to replace "uncool" parents with "cool" ones. Now that's an asshole-ish thing to do with your parents because they are not "cool", right? Later I remembered that just about every replacement made ended up in An Aesop that leads to Todd or Riley deciding to undo the replacement. Therefore, it is easy to assume that any other kid who wanted cooler parents will most likely learn that they are no replacement for their true parents and do the same. Todd and Riley were orphans, so they got an exception.
Tasumi's armor being a Japanese Halloween costume in season two may seem like a stretch, but it makes perfect sense. We never see her costume doing anything unusual, and it can be repaired by using duct tape.
In "Heartbreak In The City", Agent K tells Riley “Conrad called me" and adds "Your... Uncle Conrad” when she realizes that Tasumi and Abbey are hearing her. Flashforward to the finale...
Okay so Conrad is Riley and Todd's uncle; I can work with that...except for one tiny detail...WHY ON EARTH DIDN'T HE ADOPT THEM HIMSELF INSTEAD OF JUST GIVING THEM REPLACEMENTS AS PARENTS?! You can legally adopt biological relatives and I'm pretty sure that orphans will be sent to close biological relatives by the legal system if they're deemed fit for adoption.
He explained that he knew nothing about parenting and thus being afraid that he's not up to the task. Also we never seen any suggestion of Ms. Fleemco existing so perhaps he preferred to give them full family and help from the shadows. It is not the soundest thing to do, but people don't always act reasonable.
He actually didn't need to adopt them. If he was their closest living relative and their parents didn't make arrangements for their care, he'd probably be awarded custody just on that. He'd have to be considered an unfit parent to not get custody. Of course even when this show was on, foster care is/was the preferred method of caring for orphaned children with no relatives rather than orphanages.
Agent K and Dick Daring were forced into marriage, most likely against their will, to raise two children they'd never met before. They seem to actually be in love and having a good time, but still.
Given the way Fleemco's replacing works (the replaced party gets to live out the life they've always wanted), it's possible that the company played match-maker for them prior to making them Todd and Riley's parents.
Defied in the case of people who do get replaced. It's shown on screen what Fleemco does to get them out of the picture and it's actually quite nice; they're placed in a situation where they get to live out their dreams (for example, a Drill Sergeant Nasty cheerleading coach being hired as a water-skiing coach).