Millimetres
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Milliliters or fluid ounces would be reasonable metric units to measure the capacity of a drinking glass. These units are commonly used for measuring liquid volumes and would provide a clear indication of the glass's capacity.
Thickness is measured in units of length. You'd have to choose the one that gives a reasonable number corresponding to whatever you're measuring the thickness of. If it's the thickness of a hair, you'd probably select a sub-multiple of a meter. If it's the thickness of the atmosphere, you might select the kilometer. If it's the thickness of country gravy, that's a whole different story.
There is only one metric system. The relevant units, within that system would be millimetres.
Thickness is typically measured in units such as millimeters (mm), inches (in), or micrometers (µm).
You would use millimeters (mm) to measure the thickness of a house key, as it is a small unit appropriate for measuring such dimensions.