Is it fair to say that the Free Cities of present time has a similar crabs-in-a-bucket mentality as the Seven Kingdoms had at the time of the Great Game?

Hell yes. 

Volantis tries to conquer the Free Cities? Tyrosh, Pentos, Lyr, Myr, and Braavos unite to stop them.

Lys, Myr, and Tyrosh merge into the Kingdom of the Three Daughters? Pentos, Braavos, and Lorath ally to smash them. 

And so on.