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.