The short answer is no. It has no effect.
For example, if you stack Awakened Multistrike Support with Multistrike Support, only the power one has effect.
PoE Awakened Gem is a subtype of support gems. They are better versions of existing support gems. Its level one version is more powerful than the level 20 version of the one It replaces. Their effects do not stack with their normal versions.
In addition, they have a higher starting level requirement and require more experience to level up.