Awaken the Ancient can enchant any land with the subtype Mountain, not just one named Mountain.

The enchanted Mountain retains any other types or subtypes it may have. You can still tap it to add {R}. It can be affected by anything that affects either a land or a creature.

This doesn't count as a creature entering the battlefield. The Mountain was already on the battlefield; it only changed types.

If the enchanted Mountain stops being a Mountain for any reason, Awaken the Ancient will be put into its owner's graveyard as a state-based action.

Copies of the enchanted Mountain won't be 7/7 red Giant creatures with haste. For example, if Awaken the Ancient is enchanting a basic Mountain, a copy of that creature would just be a basic Mountain.