I was thinking about just tying one side of 5 H-bridges together, but is it really that simple?
Nah, not really. Motors under different conditions (stalled/moving) will behave differently, and thus, this will possibly have pretty adverse effects.
Aside from it sounding a mechanically bad idea to have a complex finger system, and then just move them all at once, at the same speed, in the same direction. If you wanted that, a simpler three-pronged, single-actuator-driven gripper like a grab-exvacator has would have done the job much cheaper, faster, reliably…
No, if you have a hand, you'll need to drive these motors separately, especially if you want to take advantage of the fact that a hand can actually effect different forces with different fingers, and can thus both throw stones and lift dry leaves.
So, having an MCU with too little pins to drive all the motors you have is a bad start. This should be the easy part of your hand's design! You shouldn't be making your system worse, just because you chose the wrong MCU. Fix the MCU choice instead.
Get flexible motor drivers – we can't recommend anything without even knowing the types of motors your dealing with. Get an MCU that is made for such tasks. There's a lot of MCUs that come with integrated motor control logic and high IO pin counts. It's not rare to see a 144 pin MCU, or even higher BGA pad counts!
From an ease of programming and computational power point of view, you're probably looking for something with an ARM Cortex-M4 core and quite some RAM. ST Micro makes interesting ICs for that purpose, and so do NXP and TI.