Ethnicity is pretty transient.
Are you from Nigeria? I have friends whose ancestors were Fulani traders 200 years ago but are now Edo speakers. Hundreds of thousands of Igbos were sold to ethnic groups in the Niger Delta (see Jaja of Opobo) but now speak Ijaw, Ogoni etc
Nigeria is a very diverse place. I know more about Nigeria’s history than you.
Yes, I am from Nigeria and please don't presume to know more about Nigerian (particularly Igbo) history than me.
Yes, people migrate, intermarry and even conquer other people’s lands and become something else altogether. It's the same everywhere, not just in Africa.
However, the point you made was that your ancestors were Bini and then became Igbo just because they moved Eastwards. That makes no sense. If your ancestors were truly Bini, there would be traces of it in your dialect.
All Igbos are mixed with something. In fact, the only way we classify Igbo heritage is through language. There is no other unifying factor.
So what linguistic similarities does your dialect have with Bini? Because even in some parts of Rivers (it might even be Opobo) where Igbo women married their men en masse, they still have some traces of the Igbo language in their dialect.
Where are the Bini traces in yours?
(Mind you, my ancestors also migrated from the Benin Empire but they were always Igbo as they always spoke Igbo.)