I believe the answer to your question is that the structure is "connected" to form a solid piece of material, but like any real material it isn't perfectly rigid. So it can vibrate, kind of like a tuning fork, and the thinner parts can act as springs. See the diagrams here: https://pmc.ncbi.nlm.nih.gov/articles/PMC9571586/

If you wanted to use such a device as a simple oscillator, you just set it vibrating and measure the oscillation frequency. Since the material isn't perfectly elastic, the vibrations will eventually be damped and die out (I guess this is what you mean by "friction"), so you use a feedback mechanism to keep the oscillation going.

A gyroscope works the same way, except that you use feedback to maintain an oscillation in one axis, and measure how much of that oscillation couples to a different axis. Whatever effect the damping has on the final measurement, I would assume it can be easily calibrated out.

I don't really know how they physically manufacture these devices without damaging the tiny, delicate structures. It's kind of mindblowing that they're so incredibly cheap.