The Adopted Family For the Last of their Kind
Tigger is forced to confront the Fridge Sadness of being the only Tigger in existence. Furthermore, some people take complaint with his angry reaction to Pooh and his friends as being contrived and/or only existing for the Liar Revealed plotline. To some extent, this criticism has merit, but in universe Tigger's reaction and desire for a family is very understandable.
For all of Tigger's life, he has been living with people who are different from himself. While the other characters don't seem to mind, they don't seem to be reminded of it as often as Tigger does when his bouncing antics are somewhat exclusive, especially when Roo is not around to bounce with him. Furthermore, as the last of his kind, his desire for a family is adopted from an unspoken assumption that only other Tiggers would be able to understand what it means to be a Tigger. Instead of having to announce what it means to be a Tigger, it would just be status quo. In a sense, Tigger wants to feel validated that there are other people that act like himself and that he isn't just some sort of outcast who doesn't belong. When Pooh and the friends pretend to be Tigger, it comes across as a mean-spirited prank that implies that Tigger is a clown that people can pretend to be from time to time. And Tigger doesn't want to be a caricature. He wants to be a *character*. If he doesn't have any other Tigger's, who is he supposed to empathize with?
It's only when he is forced to confront the reality that his friends are willing to help him, even to the point of chasing him through a snowstorm, while the supposed Tigger family is nowhere to be found. For a real-world example, this scenario would consist of a child recognizing that their real family is not coming back for them and/or that they won't ever find their real family because that doesn't exist for them anymore. Tigger most likely has to recognize the later and it's only when his friends reveal that his real family is made up of the friends that care for him unconditionally that he recognizes that he is home. This is all they can offer Tigger who by this point reconciles that this is the best he's got and that there is no point trying to look a gift horse in the mouth. He might be the only Tigger, but his friends still accept him anyway.
From One Bounce to Another
In addition to Tigger's Character Development, we also get Roo's. Previously a mere bit-player compared to the rest of the Pooh ensemble, Roo never got a proper character study like the others did in the New Adventures of Winnie the Pooh series or Pooh's Grand Adventure (for the majority of them, he was completely absent in fact). By contrast Roo's Big Brother Worship of Tigger is put to the full test here, with Roo showing investment in helping Tigger and gaining his approval, only to show extreme guilt and sorrow when he hurts Tigger, acknowledging that he is a person with his own vulnerabilities and desires. Roo fails to replicate and impress Tigger throughout the movie until he shows his sincere devotion to him, saving Tigger after his Heroic Sacrifice and finally pulling off the Whoop-de-Dooper Loop-de-Looper Alley-Ooper Bounce (similarly vindicating what some fans believed to be a rather cliche climax).
Tigger, who started the movie complaining he didn't get center stage enough mind you, showing his love and approval for Roo at long last and handing him his Tigger family locket at the end of the movie acts as a fitting Passing the Torch franchise-wide, as Roo would continue posing as an Ascended Fanboy for the series long after, essentially becoming the Breakout Character just as Tigger was before, making up for a long tenure of being ignored and unrealised in and out-universe.
