...how would you characterise dominate person cast with indefinite duration?..
...upcasting dominate person at eighth level can maintain up to eight hours with concentration, so i'd be inclined to consider twenty-four hours duration equivalent to a ninth-level spell, but perhaps i'm under-gauging the difference in power between eighth and ninth-level spells?..
...regardless, in this particular case it was cast with indefinite duration plus disadvantage on the saving throw against a hostile player character, which i'd like to translate into mechanical terms: maybe a ninth-level casting with heightened metamagic plus silvery barbs?..
...i'm playing with a DM who has a poor grasp of fifth-edition's mechanical balance and i'd like to help him understand just how adversarial his homebrew and houserule tendencies are leaning, especially for early tier-two gameplay...
Just to get it out of the way, I'll note that disadvantage on the save should not be possible in this situation since the PC already has advantage from the hostility, so disadvantage imposed would just cancel that out and the PC would roll normally. I get that your DM is already fiat-ing the spell's duration, but I don't know that this part was ignored intentionally.
So... when done by a DM, I'd compare this to a ghost's possession ability. Both cause an indefinite loss of control for the PC. Ghost is a CR4 monster, so it's within the realm of a tier 2 party to face. So, similar to the Ghost's possession, there needs to be other ways for the effect to end.
If the DM is using the spell as written (other than the duration), there are ways of ending it — even for a tier 2 party. There's the rather obvious "do damage to them so they repeat the save", there's Dispel Magic, and anything that will end the charm condition (e.g Calm Emotions will suppress the effect while active, or you could find/pay someone to cast Greater Restoration on them to end the effect entirely).
It is quite an adversarial thing to fiat an indefinite Dominate, but it's not mechanically outside a DM's toolkit or beyond a party's capability to deal with is how I would describe it.