"Green Man" can have a number of different meanings. In Celtic myth, it's often a representation of the horned God, and is an earth spirit associated with nature, death and rebirth. The cycle of the seasons and the return of life to the land in the spring are often associated with him. Although the Green Man is often referred to as a singular being, a deity who is repeatedly resurrected, this is not the only reference to the Green Man.
There are references to the Green Man (or men) in many, many cultures. Indeed, the Green Man is often considered an archetype, reflected in myriad stories and legends. Oddly enough, while most pagan imagery was avoided by the early Christians, the green man's distinctive foliate face appears in churches and manuscripts frequently. Possibly due to his association with resurrection? Many references also treat the Green Man as an aspect, and many beings associated with the natural world are referred to "Green Men". Basically, a slightly older version of "Tree Hugger".
Remember that Patty often plays with archetypes. The Fae all have to call themselves something, and many of them are unique beings. Beings who call themselves "Snow Elf" or "Gremlin" may not actually be anything of the sort. So, Uncle Mike is labeled "Green Man", and he's indeed Irish. This isn't intended to specify exactly what he is, or what powers he may have. It's simply a clue that his true nature is some sort of earth spirit.

EDIT: Grrrr! This was written by Mike, not Patty. My computer is dead (no room to set it up in our current little hovel) and I can't manage to log out on her system. Sorry for any confusion! I usually don't impersonate my wife . . .