Castle in the Sky: The Dola Gang of sky pirates. Even if most of them are actually related. They even call their boss "mom".
The movie version of Howl's Moving Castle has Howl refer to the castle gang explicitly as his "little family". Howl, Calcifer, and Markl have true companions-like relationship going from the start, with the orphaned Markl having Howl as a kind of surrogate father; Sophie, Heen, Turniphead, and the Witch of the Waste all join later. Turniphead leaves when he becomes a Prince again.
Manfred, Diego and Sid in Ice Age. Circumstances bring them together despite the fact that they can't get along, what with Manfred distrusting Diego (with good reason, for he turns out to be The Mole) and Sid annoying Manfred and Diego. This dialogue puts it best:
Diego: Why did you do that? You could've died trying to save me. Manfred: That's what you do in a herd. You look out for each other. Sid: I don't know about you guys, but we are the weirdest herd I've ever seen.
In The Land Before Time, it is repeatedly stated that each species of dinosaur doesn't get along with the others (even amongst herbivores). When Littlefoot's gang gets together, even The Narrator remarks "there had never been such a herd before". However, all feelings of xenophobia amongst the adults seemed to vanish once they got to the Great Valley. This is explained in the 7th movie. While the kids were off doing their thing, the adults decided to band together too for safety. Even then, they still argue a lot more than the kids do.
Charlie and Itchy in All Dogs Go to Heaven. Sure, they fight, but in the end they are always good buddies again.
Andy's toys from the Toy Story series. This is very apparent in the third film; in fact, it's practically the theme of the whole movie. It is poignant in the themes "You've Got A Friend In Me" and now, from Toy Story 3, "We Belong Together".
The Hawaiian term for this is "'ohana", as anyone who has seen the movie Lilo & Stitch should know:
"'Ohana means family and family means nobody gets left behind. Or forgotten."
Timon, Pumbaa and Simba are such in The Lion King. Even though Timon was angry at Simba for ditching them to be with Nala, he and Pumbaa still followed him back to the Pride Lands, and, upon seeing the desolate wasteland and asking if Simba was sure he wanted to fight his uncle for this, tells Simba that if it's important to him, they're in.
Oliver & Company: Oliver manages to forge a tightly-knit group in the form of the dogs Dodger, Rita, Francis, Einstein, and Tito, as well as his human owner, Jenny, and the dogs' human leader, Fagin. Dodger himself says to Oliver that the gang is like family.
The Brave Little Toaster and his gang of household appliances stick together no matter what, and it's a good thing too considering what they have to go through.
The Furious Five in Kung Fu Panda all fight alongside each other and trust each other with their lives. Though they get off to a rocky start with Po, he ends up joining them and fits in nicely with their true companions.
Sinbad's crew in Sinbad: Legend of the Seven Seas act very much like a big, loyal family who all know each others' quirks and eccentricities. A major point of the movie is Sinbad's divided loyalties between his current true companions and his old best friend, Proteus.
Max, PJ, and Bobby in An Extremely Goofy Movie. Max and PJ had been best friends since middle school, and apparently decided to let Bobby join them on or near their level sometime during high school (Bobby was more distant in the first movie). They all go to college together, rooming together, joining the X-Games Tournament together, and sharing at least one class. When Bradley offers to put Max and only Max on his team, he declines, saying it's "all of us or none of me." In the subsequent fight with the Gammas, PJ stands up to Tank to protect Bobby without hesitating, and when Tank is victimizing him, Max shouts at him, "Hey! Leave him alone!"
In Mulan, Yao, Ling, and Chien-Po are true companions with the heroine. Though they take a while to warm to her (or rather, her male persona, "Ping"), especially Yao and Ling, they become friendly with "him" after "he" completes the camp-stumping challenge. They worry when "he" is injured and attempt to stop the execution when her sex is revealed. Even when Shang no longer trusts her, they go off to follow her plan without hesitation.