Casting dominate person on a player at all sucks. Especially in DND. At least in CofD and other systems you get XP when you get fucked like that. What are you supposed to do in DND ? Sit on your hands for weeks?

I'd just tell the DM it's not fun and I don't care if the rules technically allow it. We're playing a game not running a simulation.

DMs don't have to follow the rules, they can make enemies' stuff work however they want. But if your DM made a player save with disadvantage against an infinite dominate effect, they are either an asshole, an idiot, or both.

Edit: on that note, they can set the DC for an effect however they want.

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.

how would you characterise dominate person cast with indefinite duration?

As "your character is now mine until the spell is broken, here is a new sheet for you". Reread the spell and if I would be the recipient of it I would be very uncomfortable. Agreeing with jjjalljs that when it is used on a PC is really sucks.

upcasting dominate [...] difference in power between [...] heightened metamagic plus [...]

Mechanical stuff. Permanent Dominate Person feels like the realm of a Wish spell so saying it is the equivalent of a ninth level spell sounds appropriate. Unless it was cast in combat, in front of the players and their characters I wouldn't really bother with exact rolling mechanics. Things happening behind the scenes I just let happen. But if it was in front of them, with the saving throw called for it was in front them, I'd say it would and should be fair to ask the DM what powers were behind the roll. Especially since it is practically a save-or-die (-for-a-while) spell.

Rant about enchantment spells
I personally really don't like many enchantment spells, particularly those about mind control and domination. They make me uncomfortable. When DM-ing I wouldn't use any unless I would have gotten explicit permission from the player that it would be OK for them and general permission from the table at large. They can be triggering and I always have "Mind Control/Alteration" as a topic on my Lines&Veils document I ask my players to fill in. Play safe, take care of eachother and push boundaries only with consent in safe spaces.
End rant.

You are using ninth level spell ? And this is the biggest question to ask ? This is like asking for a jumbo jet to get a light to read a book