There are many different species both similar and different that evolved in many different parts of the world. Animals have evolved all over the world and continue to do so. Although evolution was not initially started in different parts of the world, it has continued to progress as animals and plants find new areas to live in.
Yes, there is evidence that many species that live in different habitats on complete different places on Earth, have evolved many of the same characteristics and adaptations to survive and thrive in their habitat. Look at the different types of animals that feed on ants, and the adaptations that they have.
Bones of human predecessors have been found in many countries, and especially Iran, that prove that human life did not all originate in one continent. The build-up of societies in many countries inhibit the ability of archaeologists to dig up ancient bones that would further prove this theory to be correct.
My understanding of evolution includes the fact that there was once a super continent where most species started out and then they were slowly separated over time as these continents divided and moved away from each other. When these populations were broke apart they began to evolve differently leading to a variety of different species across different parts of the world. I am not familiar with the stated theory.
Multiple species of most land animals developed in one place or another and then migrated outward or became separated due to tectonic plate movement. Humans originated in Africa more than 2 million years ago and then migrated to other parts of the world. Then humans evolved into different species before one dominated the landscape.