Say I had a character that is a Brawler 6/Fighter (Varisian Free-Style) 6.
Both classes bestow Martial Flexibility, and I've been told that since there is no language about them stacking, they would be separate pools.
So, combat begins, I dip into my Brawler pool and grab two combat feats as a move action. Can I then use my remaining Standard action (as a Move action) to dip into my Free-Styler's (separate) pool of Martial Flexibility and grab two more feats?
You can replace standard action with move action.
I not sure about the stacking, but as far as i know, there is no rule said that it will limit what you can do for your extra move action.
Each one of the following classes lists two classes that it draws upon to form the basis of its theme. While a character can multiclass with these parent classes, this usually results in redundant abilities. Such abilities don’t stack unless specified. If a class feature allows the character to make a one-time choice (such as a bloodline), that choice must match similar choices made by the parent classes and vice-versa (such as selecting the same bloodline). The new classes presented here are all hybrids of two existing core or base classes.
Yes on everything.
"You can take a move action in place of a standard action." CRB pg. 181
Nothing limits this move action. A 1st level Brawler could theoretically activate Martial Flexibility twice on the same turn if they wanted to (the second feat would replace the first one, though). A 7th level Bard could start a performance, cast Reviving Finale, and start a new performance. And so on...
And yes, with no language to the contrary, your two version of Martial Flexibility are fully seperate.