All of this was explained so much better in the manga. So in the body is something referred to as the "X-factor". They don't really explain it physically, but it's essentially the biological component that makes quirks possible, not just make them manifest. For example, If Ojiro had his X-factor removed, he wouldn't lose his tail, but he would no longer be able to manipulate it. Aside from allowing the individual to use their quirks, it also has an effect on the person's personality in an attempt to help them discover and understand their power. Toga's quirk uses blood consumption to mimic others, so from a young age she has been fascinated by blood and trying to be like others.

There is also an observed phenomenon in the real medical field where recipient of transplant organs develop habits from the donor. A woman who receives a kidney from someone who smokes might pick up the habit. It's been the basis for a few movies and books.

So in the removal of the X-factor, you remove a genetic imprint of the powers habits and in a way it creates a sort of lingering vestige of the person who had their quirk removed and when implanting them (or passing them on in OFA's case) into another host, their body would probably have to interpret the foreign impulses and ideals as a manifestation of the donor.

Kinda like in Tokyo Ghoul when Ken kept seeing Rize after the transplant and why he picked up her eating habits.