The characters in Kingdom Hearts are in three categories: Disney, Squaresoft and Original/Other. Typically characters are characteristically represented depending on their categorical determination. A list of early characters as seen in Jiminy's Journal when first leaving Traverse Town is:
Donald Duck (1934)
Minnie Mouse (1928)
Daisy Duck (1937)
Jiminy Cricket (1940)
Leon [Squall Leonhart] (Final Fantasy VIII)
Yuffie (Final Fantasy VII)
Aeris (Final Fantasy VII)
Sid (Final Fantasy VII)
Tidus (FInal Fantasy X)
Selphie (Final Fantasy XIII)
Wakka (Final Fantasy X)
Moogle (Final Fantasy III)
The first type of characters are those introduced for the purpose of the game. The player and non-player characters (other than standard enemeies) are extremely limited being Sora, Riku, Kairi and eventually Ansem. These introductions are generally explanative as they are new characters, but they are not startling in most ways (Ansem might be an exception).
All of the Disney Characters are shown in the journal with their first Disney animated short appearance. All of the Disney characters are represented in as faithful a rendition as possible. The proportions are accurate and the voices are the same as with the shows (Donald and Goofy sound like they do in the cartoons and movies, Chip and Dale get their original, non-linguistic voices and not their Rescue Ranger voices). The introduction to these characters is generally uneventful and often does not even use their name or dwell on them (Chip and Dale, the Brooms). Donald and Goofy are both introduced onto the screen after a short pan, but the rest of the characters are almost expected to appear: all of the characters from Alice in Wonderland show up eventually as do the rest of the individual worlds' Disney characters. Perhaps the only character with a more forceful introduction is King Mickey when he eventually gets introduced in person during the closing scene.
Finally are the Squresoft characters. Actually, closer would be saying Final Fantasy characters. While Squaresoft had access to a host of characters from various popular franchises including Seiken Densetsu/Secret of Mana, Chrono Trigger, SaGa, Parasite Eve and many non-series games (and this expanded when Squaresoft became SquareEnix between the first and second games, and even more at present with the inclusion of Eidos), the only characters included in the Kingdom Hearts games have been from the Final Fantasy series. These include requisite Moogles (given no introduction) and Sid (given a longer introduction as the shopkeeper) it also includes fan favorites given dramatic entrances (Leon/Squall, Aeris, Yuffie and of course Cloud and Sephiroth later on). These characters are created to mostly resemble the original representations where possible (pre FFVII characters such as Setzer from FFVI seen in Kingdom Hearts II is an example of necessary adaptation due to graphical differences). However, it is only the Final Fantasy characters that are given any sort of modification such as age alteration. Selphie, Wakka and Tidus are all altered fron their original forms into children, similarly, Seifer and his two lackeys Fuu and Rai (from Fujin and Raijin) and the from Final Fantasy X-2 are turned into teens in Kingdom Hearts II.
Beyond the simple representation difference, where the Final Fantasy characters have a broader range of accuracies, the Squaresoft characters generally all have sudden and surprising entrances. Part of the reason is that unlike the Disney characters, where it is the world itself that is a surprise (which Disney movies/worlds will be visited) and the characters naturally follow, the Squaresoft characters are essentially random, based in fan appeal and meant to surprise and satisfy the fans. Therefore, their entrances and attitudes are completely different from the Disney characters.
On another level, however, there is also the question of universalization of Disney and localization of Squaresoft. This answer might be more intersting and worth dealving into in the future.