I do not want to remove the Legacy on my pets profiles accidentally so I came here to ask.
What will remove the (Legacy) under a Legacy pet on their profile? The only way I've heard of is to adopt or transfer the pet.
Example of a Legacy if you haven't met one before: HaneKitsune
The only thing that will remove the (Legacy) is if you adopt/transfer out the pet. So long as it remains on your account it will be fine.

Oh cool! I didn't even know this was a thing. I have several legacy pets. :)
[tot=Barghest] [egg=Barghest] [tp=Barghest] [flower=Barghest]
Thank you. I was thinking of a name transfer and I didn't think it would be a problem but I wanted to be sure. :)
No problem! So long as the original pet itself stays on your account you can rename it or name swap it as much as you want. The Legacy status is attached to the pet ID number, not the name. :)

If I remember correctly, "legacy" simply means that the pet was already on your account on the day they added adoption dates to a pet's profile. Since they didn't keep track of when pets were added to accounts before then, they just made the birthday and adoption day the same and added "legacy" to mean the pet has been there at least as long as adoption days were implemented.