There are quite a few misconceptions generated about this topic, in my opinion. Firstly, the personality doesn't reincarnate. it is just the soul, and the eternal Self within the soul that reincarnates.
We are a series of bodies, or sheaths. The outermost is the physical body. That's all it is ... the physical body. Next inside that is the intellectual/ego/astral body. That's where we dream from. Its basically an etheric copy of our physical body. It contains this personality, this ego.
Yet further inside is a soul body. Its a body of light, humanoid in shape, but more like those guys from Close encounters. From this body intuition and direct cognition operate. This is all that reincarnates, firstly into a new physical body, inside the womb at some point in time. The astral or intellectual body grows as as the physical body grows. Little children don't have all that congested intellect, emotional insecurity and all that we adults have. The new intellectual body and ego has to grow along.
The physical body and the astral body do not reincarnate. So anyone hoping they (they meaning the ego/personality) will come back are SOL. But one of the primary differences between east and west is who we are, or better, who we perceive we are. In the East, we are souls with physical bodies. In the west we're physical bodies or egos with souls. Eastern people tend to view life as a string of lives.
When we die, we at first enter into the second world, or astral plane, astral body intact, but within time, it too drops off, memories fade, attachments fade, and once again we are just the soul, and the Self (essence) within that. Then we are ready to pick out or have chosen for us another physical body.
We could be so clouded as to choose a mammal, but that's as far as that one goes. Insects, birds, vegetables, are all different vibrations, call them souls if you wish, but these souls we are cannot use those physicals as temporal homes. Where that myth got started, I don't know. A lot of hearsay gets started by others critical of the faith in reincarnation in an attempt top make it appear less logical.