Not that we've seen
If you recall back to the original Thor movie, the enchantment was placed specifically on Mjolnir during the course of the movie by Odin, as part of his attempt to teach Thor humility and responsibility. The fact that both Thor and Loki attempted to lift it in Thor and were surprised to have failed also implies that the lifting aspect was specifically tied to that worthiness enchantment. Also, as we learn in Thor: Ragnarok, Hela once wielded the hammer herself and there appeared to have been no such worthiness enchantment in place at the time.
So as far as we've seen, Stormbreaker is just a normal weapon in that regard. Theoretically it should be "wieldable" by anyone, though effectively used by Thor alone, given his powers. It's also worth noting that it does have some enchantments itself, such as the
Theoretical/proven Bifrost powers
but that was a stated enchantment, unlike the worthiness one.
Could it have such an enchantment? It's certainly possible, but there's no evidence that would allow us to conclude such. If anything, Groot's ability to lift it could be taken as additional evidence against.