Do all Aniimo reach the same max stats?

Rarity decides where an Aniimo starts, not where it finishes. Standard attributes cap at a combined total of 20 split between Initial Potential and Acquired Potential, so a rare catch simply arrives with more of that 20 already filled in. Everything left over is Acquired Potential, and you raise it yourself with Star Essence and Star Sands, which feed both stats and potential.

The practical effect is that a common Aniimo is a materials problem, not a dead end. If you like a creature, keeping it is a real choice rather than a sentimental one, because the ceiling it will hit is the same ceiling the flashy catch hits. What differs is how much you have to pour into it to get there.

You cannot read any of this off the creature by looking at it. A captured Aniimo has to be scanned at an outpost identification station before its detailed stats are visible, so the appraisal comes before any decision about what to invest in.

What still separates one Aniimo from another

Stats converge. Traversal does not. Certain blocked or inaccessible locations need a specific Aniimo's traversal ability to reach at all, and no amount of Star Sands turns one creature into another. So your team choices still matter, just for movement and access rather than for raw numbers at the top end.

Not settled: How much Star Essence and Star Sands it takes to close the gap from a low Initial Potential to the full 20 is not settled, so the cost of levelling a common catch to the cap is still an open question.