I have been watching the later years of the series and it appears that Alex came to live with them but was never adopted and I don't believe Ann ever became his legal guardian.

I know Alex's father died and he refused to live with his mother

I am asking this question because there is an episode that revolves around Alex getting a spanking and I don't think that should have even been brought up if she is not legally his guardian.

I have watched every episode of the show but don't remember if she adopted him or became a legal guardian.

Alex had the spanking coming even though he was not legally adopted. Ann was his guardian and he was to follow Ann's rules. I remember that ep. As to soon as Ann hit Alex's butt, the picture went to freeze frame with that fake applause to help close out the show. LOL.