1 - "Anniversary" does not have an explicit earth-centric origin. It comes from Latin annum, which means year, and is thus generic.
2 - There appears to be no generally agreed-upon term that would encapsulate anniversary with respect to the Martian year. See the article Timekeeping on Mars for some background. I tend to think that Mars anniversary fits the bill, but if you were writing a science-fiction story you might want to coin a term in order to achieve some kind of verisimilitude. @PeterShor's "Martianniversary" in question comments is very interesting in this regard.
Some astronomical terms can be body- or body-type specific. For example, perigee is the closest approach of a body to earth (geos), whereas perihelion is the closest approach of a body to the sun (helios). On the other hand, periapsis is the generic term for closest approach of one body to another. See Apsis on Wikipedia.
Incidentally, I have a suggested coinage for Martian anniversary that I just now coined:
Mars in the Greek language is Ares, making the term for the closest approach to Mars of the moon Deimos, for example, into periareion. By this token, a Martian year could be called an Areannum, making a Martian anniversary an areanniversary.
You heard it here first, folks!