datreveno

A compound that explains itself the moment you split it: dato plus reveno, the return of a date. That is a better description of what an anniversary actually is than the English word manages, since anniversary only tells you that a year is involved and leaves the recurring bit implied.

The birthday question needs an honest answer. Naskiĝtago is the unambiguous word for a birthday: birth-day, exactly as English builds it. Datreveno is the general anniversary, and it is used for birthdays constantly, especially in the possessive (mia datreveno) and in the polite register, roughly the way an English speaker might say it is my special day rather than naming it. Both are correct; if the context does not make the occasion obvious, naskiĝtago is clearer, and if you are writing a card, feliĉan datrevenon is entirely idiomatic.

Where it earns its place is the non-birthday anniversary, and Esperanto has a lot of those. Movement history is measured in datrevenoj: of the first book, of the first congress, of a society's founding. Centa datreveno is a centenary. And because the compound is transparent, you can build the specific one whenever you need it and nobody will blink.