Not gears per se, but yes we measure internal bores routinely to less than .001" tolerances, and use bore mic's similar to the ones you showed. There are more precise, higher-resolution instruments (google air gages) but they are spendier and cover a narrower range of sizes per gage. There are also less pricier bore gages (2-pt internal micrometers or split ball and indicator types) that can be nearly as accurate as the hole mics, they also need more setup and pre- and post-calibrations to ensure the accuracy of the measurements, and can be trickier to use, and can give false readings for roundness. I have had people tell me that good cmm's can probe i.d.'s to better than .0001" resolution, but have some issues with those statements, ymmv, etc.
Many years ago, in an automotive gearbox plant, gear bores (ground) were checked with two-point air gauging and also roundness checked on a Talyrond roundness checking machine. Just in case the geometry was a lobed equi-diameter figure. Shafts (some centreless ground) were also checked.
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Usually "really precise" implies removing the human factor as a source of measurement error and using a CMM, and yes, I've had many things CMM'ed including gears.
We use air gaging on all the final ID/OD measurements for our bearings. Roundness and other measurements use different gaging but diameters are covered very well with air gaging. As btrueblood mentioned, you do need a lot of capital to measure a wide range of sizes though.
So what type of gear? An internal ring gear? An external gear like planet pinions or such? Tapered gear, like bevel gears?
Hypoid gears? Picture please. A high quality CMM should work just fine for measuring them.
