Melee +X can apply to any melee attack, including counterstrikes, but it can only apply to one attack per attack action. You do not really get a choice which attack it applies to either, as it has to apply to the first possible one, but the first possible one is not always the first attack in the sequence.
For example, you could use the Hydra's triple strike and wait until after the first attack finishes before revealing Bear Strength. This would then grant the Melee +2 bonus to your second attack.
Similarly, if you attack an uninjured creature with [mwcard=MW1C17]Goran, Werewolf Pet's[/mwcard] doublestrike then the first attack would not get the Bloodthirsty bonus, but the second one would (assuming the first does damage).
I know this isn't something you asked about, Boocheck already answered your primary question, but it is a subtle rule often overlooked that I felt pertained to your inquiry.