A solid slab of crystal naturally oscillates at frequencies at which one of its three dimensions, length, breadth and thickness, is a mechanical
1/2-wavelength. It can easily be induced to oscillate at harmonics of the fundamental.
It can also oscillate in one of several mechanical modes, eg., longitudinal, breadth-wise or in torsion. And in shunt or series-resonant electrical modes.
The circuit it is embedded in can encourage a preferred frequency. It is easy to select harmonics. Self-preference is also given to the frequency which has the highest Q, ie., the least mechanical loss. This is usually the fundamental.
It does not oscillate EXACTLY at multiples simply because it has three dimensions and Length, Breadth and Thickness slightly 'interfere' with each other.
A poorly cut crystal, eg., lack of parallelism, at which there may be no strong preference may jump erratically between two non-harmonically related frequencies.
Frequency versus temperature curves depend on oscillation mode and on the angle at which the slab is cut relative to the direction of the individual crystals in the bulk material lattice as found by optical means. Cubic curves are best because they contain a flat horizontal portion.