Does refinancing a house to be in my name only equal buying a house?

No. But you don't say exactly "what" was changed to be in your name only. The mortgage? The property deed? Both?

Refinancing also does not change your cost basis in the property in most cases. However, without knowing your relationship of the other person that used to be on the mortgage, deed, or both and without knowing what state you are in, it's impossible to tell.

For example, was the refi the result of a divorce where you bought out the ex and refi'd to get their name off the mortgage and the deed?  If yes, then you retain the original cost basis of whatever was paid for the house when it was purchased by both of you originally. It does not matter how much you paid the ex either.