My parents split when I was 16. After 26 years of marriage. It was mutual, but I couldn't really comprehend it at the time. They raised us with such a strong sense of family, then one day it all just seemed to end.
Both of them eventually remarried, and both demanded the same honor to their new spouses that you describe. And both became very nasty towards the other. Each wanted me to " choose sides."
I politely refused both requests. And just continued loving both of my parents (not so much the new spouses).
10 years later each of my parents expressed (independently) their regret of their devorce.
This is a long post to say I know exactly how you feel. Your Dad's desire to have you affirm his new wife is natural, and it is a sign of how much he respects you and needs your affection and affirmation.
You don't owe any respect to his new wife until she earns it and she certainly doesn't demand the same respect as your mother.
But Don't forsake your dad. Talk to him. Work out you feelings. This doesn't sound like a sin to me, only something you may not like ...