I was asked this question in a comment and thought I would answer it now, although I believe that every family treats the day in different ways, and I don't think there's any one 'right' way.
We look at adoption anniversaries in a way that is similar to wedding anniversaries: it is the day we became a family. So, we have tended to celebrate that a new person joined our family simply by celebrating who they are. On Song's anniversary, we usually eat out Chinese food, sometimes with just our family, other times with extended family. Because there are no Russian restaurants around, we usually let Alex choose a restaurant or place. Occasionally, he chooses his Nini's house! I don't know what we will do for Ella's first anniversary! I suppose I could whip up some cow tongue soup like we were served in Colombia. Ugh!!
We haven't done gifts, partially because I didn't like how that might seem to the bio kids, and also because I wanted the focus to be on the child and not on gifts. To that end, everyone shares something they like about the anniversary child. It doesn't need to be profound, but I think we all like to hear the things that others love about us.
So, all in all we're pretty low-key but we would never dream of NOT celebrating the days when our family became Russian/Chinese/Colombian!