is a square. How many squares in the same plane have two or more vertices in the set ​?

Choosing 2 adjacent vertices gets us 2 squares.  One is the original square ABCD.

There are 4 ways to choose adjacent vertices and this gets us a total of 5 unique squares.

Choosing diagonal vertices gets 2 squares each.  There are 2 pairs of diagonal vertices.

This gets us a total of 2x2=4 unique squares.

Thus there are a total of 5 + 4 = 9 unique squares, one of which is the original ABCD