The tattoo would disappear if a new one is replicated.
The magic item being replicated is "Spellwrought Tattoo", so the needle isn't relevant and a new tattoo can't coexist. But if you need more reasoning, then consider this (emphasis mine):
Produced by a special needle, this magic tattoo contains a single 1st level spell, wrought on your skin by a magic needle. To use the tattoo, you must hold the needle against your skin and speak the command word. The needle turns into ink that becomes the tattoo, which appears on the skin in whatever design you like. Once the tattoo is there, you can cast its spell, requiring no material components. The tattoo glows faintly while you cast the spell and for the spell's duration. Once the spell ends, the tattoo vanishes from your skin.
There are equivalent common item tattoos which are not consumed, for example Masquerade Tattoo (emphasis mine):
Produced by a special needle, this magic tattoo appears on your body as whatever you desire.
To attune to this item, you hold the needle to your skin where you want the tattoo to appear, pressing the needle there throughout the attunement process. 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.
So even if focusing on the needle, it can be argued that the Spellwrought Tattoo is the needle and as long as the tattoo has not been used, it should take up an infused item slot. Infusing a new one would therefore make the previous one vanish / reappear as a normal needle, disallowing stacking.
RAI obviously spell stacking is not intended because the Artificer has a cap on Infused Items and Infusions Known, going so far as to prevent using the same infusion in more than one object.