There are two processes at work here - either one of which can cause confusion.

The first is that there are some similarities that one can find throughout European folklore. Is Robin Goodfellow "often portrayed as a satyr" as you assert? I'm not sure I have seen that specific thing expressed in those precise terms, but there are similarities in that a great deal of Europe had traditions about similar forest entities - using a wide variety of terms - who were the embodiment of the vibrant life in the woods. These parallels can be account for as descending from a common ancestor or as being a natural thing that people could imagine. Regardless of the explanation for the similarities, this allowed for the diffusion of stories - because diffusing is what oral narratives do best! - and that process can cause diverse traditions to seem similar.

The second process is a recognition that Shakespeare and anyone else who could claim to be educated or semi-educated until recently would be well-schooled in Greek mythology. This caused a certain amount of intellectual diffusion/influence not unlike the folk process that had engaged and modified oral traditions throughout the region. Because people knew of Greek mythology, there was an open door to have this prestigious body of literature to work its way into indigenous works of literature. And again, the path was clear for this to occur since there were native similarities between the traditions.

This line of reasoning can be taken too far, and the impression that Greek myth is behind British folklore is certainly an illusion. Much of what occurs in "Midsummer Night's Dream" is good, old-fashioned British traditions (with a few Greek names inserted for Classical flair). One need not go too far afield to understand Shakespeare's sources as English folklorist, Katharine Briggs demonstrated in The Anatomy of Puck: An Examination of Fairy Beliefs Among Shakespeare's Contemporaries and Successors (originally 1959).