It’s up and active. There are pet battle world quests and vendors sell pets for charms. Not sure about pvp battles but in other cases, it’s not abandoned like archaeology.
Define ‘relevant’
New pets are added in, new enemies are added in. It has always been a side activity, something minor, that won’t change.
It won’t be tied to power gain, it’s basically like transmog. There for those who enjoy it, rest can ignore it.
If you want to challenge yourself in it, you can with achis or i am sure there’s a pvp community with whom you can pet duel.
Far from abandoned. Very active, with two very active websites and two very active Discords devoted to them outside the game proper.
Collecting and PvE battling are going strong. PvP, as people have said, if a bit iffy. For PvP, a lot depends on which “pet battlegroup” you are in, which decides your queue time, and the devs seem determined to make daft decisions that lead to a boring meta, so not many people active.
Dragonflight introduced a short quest chain giving you three Blue Level 25 pets, so at 70, that’s one quick way in.