When a guy refers to you as "my girl" what is he saying?

Anonymous
We've been dating about a month or so, he says he does not want a serious relationship, but will be monogamous. The other day we were lying together on his couch and he referred to me as "his girl" now how can I be his girl if he said no serious relationship, and we are open to date others.
When a guy refers to you as "my girl" what is he saying?
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