Yes and no.

If he'd gotten powers from the divine oath-giver he'd be a Warlock or Cleric, dependent upon the nature of their relationship and the being's powers.

If he got the powers himself from his absolute rigid dedication to his oath, then he'd be a 5e Paladin (I prefer "Dedicant" or "Crusader" for which Paladin should be a specific Oath but that's a different conversation).

Otherwise in older editions he'd probably just be a devout warrior.

For those older editions he'd only be a Paladin if the oath he held to was far more specific and arguably he and several of the other hobbits were a bit too quick and dirty for. Particularly during the era of Racial restrictions to classes which didn't allow halfling Paladins. (Assuming halflings to hobbits is 1:1 in all settings, which is far less consistent over time.)

For how a generous DM might work around that in older editions sometimes, I'd look to BG2's Mazzy Fentan: https://baldursgate.fandom.com/wiki/Mazzy_Fentan