If not spells, they at least have 'limit breaks' somewhat like those in Final Fantasy (and swords!). I will address them as such.
More specifically, I'd see Polly as a monk. She is designed to deal damage skillfully with her strength, whereas fighters are designed as damage absorbers.
From a RPG perspective, Speedy is set up as a character adept at fighting other major characters. He excels at precision and agility based combat, rather than Polly's brute strength. From a Final Fantasy perspective, Speedy would be somewhere between a ninja (no surprise there) and a Samurai (bet you saw that one coming). His 'limit' attacks would be a level higher than Guido's/Polly's. From a D&D perspective, Speedy would be a mix of rogue and fighter, with many ranks in parry. (WoW wise, he'd be a pre-1.10 Shaman

.)
I'll admit that Guido doesn't fall into any traditional category. He ends up as a mish mash of Speedy's roguish style, with far more invested in his personality and charisma. Very little of this comes through, as he tries to go where his stats wont support him (he fails ALL his persuade checks). I'd say that he (D&D) is a rogue that invests more of himself in Charisma, Persuade, and such skills that will further him outside of combat.
They all follow the same general path, but diverge further down the line.
As for other characters:
Bad Bird: Final Fantasy wise, he'd be a Ninja/Dark Knight. Overall, he'd be oriented as a more ruthless and evil version of Speedy (with a greater focus in Intimidation.)
Lucille: Hard one to pick, but Lucile ends up as a summoner. She can 'Summon' and incredibly powerful swarm of missles, but few times each day.
Francine: A bard, using song/poems to inspire and empower the cats.
General Catton: a fighter specializing in damage reduction and good against mobs of less powerful enemies. He also has some parts as a item based mage, using the cannonballs of his cannons to deal damage.
Meowsma: A druid/Geomancer (which I post with reluctance). He is more attuned and at home with the earth, and is able to use it. He can manipulate the landscape around him, and use the Druid's traditional cyclone.
Spritz T. Cat: A hydromancer, and the only true 'caster' of the cats.
Bat Cat: In Final Fantasy, he'd be a lancer. Uses the 'Jump' skill.
Jerry: The closest that I could get with him would be a meditator. Uses the 'Talk Skill'.
For a villain, the Seymour would be a well underpowered Kefka. Dresses funny and commands through fear and intimidation.
The Catatonic: Holy Swordsman all the way. People that have played Final Fantasy Tactics will know of what I speak.
...
So now that I've shown myself as a FF/RPG nut, I'd say that you were fairly accurate.