Undyne is 200% a Dragoon.
And Papyrus is totally a Spaghetti Knight. It's like an Onion Knight, but with Spaghetti.

Frisk: Pacifist, a mediator-like class that can boost the party morale and stats and cast debuffs on hostile enemies and convicen minor neutral enemies to flee, leaving no EXP but increasing item drop rate.

Papyrus: Skeleton Knight. Focuses on melee attacks and debuffs.

Undyne: Monster Hero. A Dragoon-like class with bonuses agains human enemies.

Toriel: Protector. Mainly a sturdy white mage with some basic offensive fire magic. Can cast Defend on one party member and shield him/her if enemy attacks would KO it. Consumable items used bu her will have double effects, both on HP restoring and buffs.

Sans: Time Prankster.A super mage with extremely powerful ???-elemental magic, speed buffs and debuffs. Insane magic and speed, awful everything else. Has a negative passive skill that negates any HP, DEF and RES gains from equipment.

Playable at the prologue:

Asriel: White Prince, literally Frisk's skillset with Toriel's healing magic and Defend.

Asgore: King, has overpowered versions of Undyne and Papyrus' melee skills and Toriel's fire magic and basic healing magic.

Chara: Kid. More or less like a FF Tonberry. Huge speed and has a "Play" command, a purposedly overpowered knife stab with very high attack, huge chance to crit and ignores defense.