You've already said it in the the title of your question.
If you don't specify anything, an anniversary is thought to be a wedding anniversary. Otherwise, you specify what it's of in the sentence:
This is the third anniversary of my promotion.
This is the fifth anniversary of our store being open.
This is the tenth anniversary of us moving here.
Note that anniversary normally applies to an annual event:
1 : the annual recurrence of a date marking a notable event
// a wedding anniversary
However, it doesn't have to. You can specify something other than an annual period of time if you use an adjective before the noun:
broadly : a date that follows such an event by a specified period of time measured in units other than years
// the 6-month anniversary of the accident