I had been thinking “one of my …” and “a … of mine” were interchangeable.
But you seem to say it isn’t. On the Internet, there are many examples of sentences using that expression.
Is this perhaps a question of "good" English, "perfect" English, etc?
It is very strange that either "one of my friends" and "a friend of mine" sounds perfectly fine, but only "one of my dogs" sounds okay and that "a dog of mine" sounds a bit odd. Not ungrammatical -- just a bit odd.