The tatoo likely sticks, but check with your DM
Find familiar (p. 272, PHB) does not say familiars drop all magic items1; it says they drop all worn or carried items:
Whenever the familiar drops to 0 Hit Points or disappears into the pocket dimension, it leaves behind in its space anything it was wearing or carrying.
The tattoo, however, is not worn or carried; it instead becomes part of the creature's skin:
Once inscribed on a creature's body, damage or injury doesn't impair the tattoo's function, even if the tattoo is defaced. When applying a magic tattoo, a creature can customize the tattoo's appearance. A magic tattoo can look like a brand, scarification, a birthmark, patterns of scale, or any other cosmetic alteration.
It is a cosmetic alteration of the skin that is part of the creature. Worn or carried objects in contrast are not part of the creature, they are separate objects. So the clause above does not say the familiar will drop the tattoo when it vanishes.
Here is what the rules say about the familiar appearing, disappearing (from the spell description, p. 272, PHB)
You gain the service of a familiar, a spirit that takes
an animal form you choose.
Disappearance of the Familiar. When the familiar drops to 0 Hit Points, it disappears. It reappears after you cast this spell again. As a Magic action, you can temporarily dismiss the familiar to a pocket dimension. Alternatively, you can dismiss it forever. As a Magic action while it is temporarily dismissed, you can cause it to reappear in an unoccupied Space within 30 feet of you.
That is all the rules we have for it. Now, spirits do not normally have a physical body, but once you have your familiar appear by taking an animal form, it does have a physical body, and the description does not say it loses this body, merely that it "disappears" (who knows where to) and you can cause it to "reappear", or that it is dismissed to a pocket dimension and can be magicked back. So technically, in never loses its form or body in those cases, and the tatoo should stay on.
Here is what it says about changing form:
One Familiar Only. You can’t have more than one familiar at a time. If you cast this spell while you have a familiar, you instead cause it to adopt a new eligible form.
When you chose a different form, it adopts this new form, but the rules do not say if that happens by becoming disembodied and creating a new form, or if it is just transforming into a new form. So this is left to the DM. I think it is more fun for the players, and more consistent with the other ways it can vanish and re-appear, if the tatoo stays on.
1Magic Tatoos in tattoo form are magic items. They are listed under "Magic Item Descriptions" on p. 119ff of Tasha's Guide to Everything, and their description clearly says they are Wondrous items, for example, for Absorbing Tattoo (p. 119)
ABSORBING TATTOO
Wondrous item (tattoo), very rare (requires
attunement)
For this, it does not matter if the tattoo is in needle form, or if it is in tattoo form. It is a magic item throughout:
When the attunement is complete, the needle turns into the ink that becomes the tattoo, which appears on the skin. If your attunement to the tattoo ends, the tattoo vanishes, and the needle reappears in your space.
The needle does not enchant the creature with a spell effect, it becomes the tattoo. So the tattoo is the magic item. Further evidence can be seen in the fact that if you end attunement, the tatttoo turns back into the needle.