MTG Arena's Jumpstart: Historic Horizons introduced a digital only keyword called Perpetual which changes a card's characteristics permanently and throughout all zones. Paper magic isn't able to do this as it requires tracking the card in every zone.
Paper magic does have cards that grant effects to players "for the rest of the game" and "effects that last indefinitely" to a permanent as long as the affected permanent is on the battlefield.
"This effect lasts indefinitely" was a piece of reminder text that began appearing on cards in the Invasion set, though quite a few earlier cards were errata'd to have this line as well. This reminder text stopped being used in cards as of the original Mirrodin block. Most likely to back up Magic's idea of playing cards as they are written.
The first card to have an ability that reads 'for the rest of the game', without a keyword such as 'Epic' or 'Ascend', was Stigma Lasher from the Eventide set.
So, yes, there are paper magic cards that can affect a permanent 'permanently' as long as it's on the battlefield, though nowadays you have to look out for them as they no longer come with reminder text.
