Most rescue owners have no idea when their animal was born. The day they came home is the date that means something, and it is the one worth keeping.
You will see it called a gotcha day. We call it an adoption day, for two reasons. "Gotcha" is what you say when you catch someone out, and nothing about a rescue is a prank. And adopted people have been saying for years that being described as got is a strange way to describe being loved. Their objection is about families, not pets, but the word does the same work in both places.
Use whichever you like. If gotcha day is what your household says, that is the right name for it.
Some people mark it quietly. Some throw a party. Both are fine.
If the shelter is still going, tell them. They rarely hear how it turned out.
A tree dated to the adoption gives the anniversary something to point at each year.