There’s a weak argument to be made for the two to stack and for the character to be able to maintain three styles, but it takes a lot of mental gymnastics and is very unlikely to have been intended. Intent doesn’t really matter all that much here, though, since the authors probably didn’t consider the combination at all and so didn’t really intend any particular thing for it.
As you note, master of many styles is explicit about not allowing you to go over two; we aren’t going to gain much from that. However, D&D 3.5e and Pathfinder have a long, long history of classes assuming you were single-classed in their write-ups. See this Q&A on 3.5e sorcerer/wizards, this Q&A on a PF scarred monk/ninja, and so on. So it’s very easy to read “[A master of many styles] may only have two style feat stances active at a time,” as “The master of many styles archetype does not give you the ability to have more than two style feat stances active at a time.” This is pretty common understanding in 3.PF and could easily apply here. So if something else were to say “you can have an additional style feat stance active at a time,” that could stack with master of many forms to get to three active style feat stances.
The question then becomes, is free style fighter that “something else” here? Unfortunately, strictly speaking, the answer is no: free style fighter gets to “mix two styles,” and gets to have “an additional style feat stance active simultaneously with his first style feat stance.” The idea that the extra stance is being added onto a singular existing stance, and that the expected result is two, is explicit in the text. We can make an argument similar to the one we made for master of many styles here, saying that this lets you get two, but something else could bring in a third if it wasn’t explicit about adding up to two.
But that leaves us with both free style fighter and master of many styles looking for “something else” that isn’t as restricted. There are some arguments we could make—you could argue that since the free style fighter is “mixing” the two stances, they then count as one and the master of many styles is then able to add another stance as the “second” and thus continue to obey master of many styles’ insistence that you cannot have more than two, but this argument is really weak when free style fighter itself refers to the stances as two rather than one.
A more fruitful argument is to say that, unlike master of many styles’ explicit insistence on a limit of two, the free style fighter’s indication of two is more casual, more just “this is the number we assume you have, because [as we’ve already discussed above], we’re writing from a perspective of assuming you are single-classed.” You could then argue that it isn’t meant to be a hard restriction, just the author’s assumptions, and that you are free to use free style fighter to add a third active style feat stance to your existing two from master of many styles. The problem with this argument is that it relies on contrasting the statements in each, which implicitly strengthens the statement in master of many styles and maybe makes it harder to buy that it was also just done assuming you were single-classed.
In the end, though, you have two abilities that are each trying to enable the character to use an additional style feat stance. It’s unlikely that the authors considered the combination explicitly—one, since they rarely do that, but more importantly two, because if they were thinking about the combination they would probably have copied the text from one to the other to keep them consistent and clearer. The fact that we have two different text blurbs for this suggests, but does not prove, that the two were written independently and without awareness of the other.
So what should the GM do with this situation? Strictly speaking, it looks rather like having both is just redundant and you get nothing for doing it—that’s pretty lame. Are style feat stances so powerful that it’s worth ruling things a bit lame in order to protect the game? Each GM is going to have to make their own judgment on that, for their own table, but for what it’s worth, if it’s me?—then no, absolutely not, style feat stances are, at best, good but not great, and this entire restriction is more than a little silly, in my opinion.
So my conclusion is, strictly speaking, it’s really hard to justify getting three active style feat stances from this combination based on the text. However, going beyond the text and thinking about what will result in the best game experience, my suggestion is absolutely that the GM should allow this combination to use three style feat stances.