No, it does not include the Strength increase
There is no indication that the Strength modifier is included already in that increase. This extra damage is most likely intended to emulate the Weapon Specialization that several other classes get. The wording of that:
You’ve learned how to inflict greater injuries with the weapons you know best. You deal 2 additional damage with weapons and unarmed attacks in which you are an expert. This damage increases to 3 if you’re a master, and to 4 if you’re legendary.
Notice how this parallels the animal companion wording. Weapon specialization isn't always paired with a modifier increase, so it can't be assumed to include that. Similarly, we shouldn't assume that it does for animal companions.
Further, when an animal companion becomes Specialized, we see this line of text:
[...] it increases its additional damage with unarmed attacks from 2 to 4 or from 3 to 6.
An animal companion that becomes Specialized doesn't always receive a Strength increase. (e.g. the Ambusher specialization). So any Strength increase should stack with the flat damage bonus to unarmed attacks granted by Savage, Nimble, etc.