no, it's not a valid target for the spell.
And what would doing that have to do with multiclassing even if it worked?
Instant enemy can let you treat a creature as the type of your favored enemy. Even if you have favored enemy (animal) though a synthesist eidolon isn't a separate creature. So no instant enemy doesn't help because it can't target the eidolon.
& no you don't retain control of awakened creatures. They stop being your animal companion if they were before, for example.
Awaken can only be cast on animal or tree. Since an Eidolon is an outsider it cannot be cast on them. Instant Enemy allows you to treat the target as if they were a type of your favored enemy. Instant Enemy is a 3rd level ranger spells, and only works for rangers or other classes that have the class feature favored enemy. Awaken is a 5th level druid spell. So even if using instant enemy allowed you to cast awaken on a creature you would need to be a 9th level ranger and a 10th level druid. Since the game only goes up to 20th level that would mean that you have 1 level left.
Also when you canst awaken on a creature it can no longer serve as an animal companion, familiar or special mount. Since the summoner came out long after the spell it stands to reason that it would no longer be an eidolon. If this did work, which it does not it would cause the summoner to lose the eidolon.
The eidolon is a class feature and as such cannot gain levels. It progresses based on the summoners level nothing else.
A spirit summoner with nature spirit could choose awaken as one of his summoner spells. Then it should "work" on the eidolon with share spells. I'm not really sure what the end result would be though.
Regardless, it wouldn't stack with synthesist.