on the same morpheme, my father - אבי, my my brother - אחי, my sister - אחותי, my grandmother - סבתי, my grandfather - סבי (though the last two considered these days a bit formal and uncommon).
Maybe אבא שלי and not אבי is a matter it respect. But i don't remember ever hearing אבי אמר...
well... I do XD but I don't have any connection with my father so... it isn't common, but now and then I still hear people saying אבי