It is possible to find someone and become attracted to in any setting. It seems that the two of you had hit it right off from the start.
You may want to set some kind of a limit (in regard to him living with his ex girlfriend)
It is unclear whether or not she still believes that they are together or she is clear that he is dating you. That would be of importance in letting you know your next step.
If he had been honest with her, then he ought to make arrangements to no longer reside with her. If he has not, that may cause you more headache in the long run (if he is not honest with his allegedly ex partner, he may at some point become dishonest with you)
BotXXXXX XXXXXne is, he is dating you now, and it is preferable that he does not reside with another woman.