You are right that the game does not require a subscription any more, so you (and anyone else with an account) can simply log in and start playing without having to put in card details or anything like that.

But there is also an optional subscription (called ESO Plus) which gives you some in-game benefits, like crowns every month and +10% XP.

If you already had a re-occuring subscription payment set up, or time remaining on a sub bought with time cards, any remaining time will be converted to an ESO Plus subscription and if you've got it set up for regular payments you will continue to be charged.

If you want to stop that happening you will need to cancel the sub. You will still be able to play the game, you just won't get the bonuses and will have to buy crowns if you want them.

And no, you can't register it to Steam. The Steam and non-Steam versions are completely separate.

You could play thru steam but hat would require you rebuying it on steam, which in sure you don't want to do. But that's the only option. Or you could just register it on steam as a "non steam game" that way your steam friends could see that you are playing it.