To start with I think the use of ‘tease’ in your sentences sounds a bit odd - I can’t quite see how ‘father’s name’ fits in. With tease you really need some word to introduce what the teasing is about. Let me give some examples:
He teases me about my appearance/my hair style/my accent. (You could also add ‘my father’s name’.)
He teases me with remarks about my appearance etc. (You could also add ‘my father’s name’.)
Or, and I think we’re now getting to what I take as the meaning of your sentence:
He teases me by calling me by my father’s name.
Hope this fits the bill.