Does the frequency of a crystal influence its height?

Influence, yes. Determine, no. For example, here's selection of shapes and sizes for the same spec crystals (and some ceramic resonators...)

The frequency is determined by the size, shape and cut of the quartz crystal inside the package, as well as where the electrodes are placed on it.

There seems to be a confusion between the size of a (quartz) crystal package, and the crystals' physical properties which include the crystal's size and it's resonant frequency due to its piezoelectric properties.

In your photo, the crystal on the left appears to be a very common HC-49/U package size. The one on the right is most likely an HC-49U/S.

From ICM, a crystal manufacturer, "The frequency is a function of the thickness of the crystal. By carefully polishing or lapping a crystal, it can be made to oscillate at any frequency." (source) In fact, amateur radio radio operators ("hams") historically were known to polish surplus crystals to their desired frequency, back when the cost of manufacturing was much higher. The practice was sometime referred to as "rock grinding." (ref one & ref two)

You can verify this by looking at a manufacturers' specifications for crystals in different packages.

E.g. FOX through-hole crystals and ICM, where in fact the larger package includes overtone crystals (page 3, PDF) for higher frequencies (looking at HC-49/U and HC-49/S). At fundamental frequencies they are often the same frequency range available.

And the size of the crystal! With the usual AT-cut crystals the size is inversely proportional to the frequency - low-frequency crystals are larger than high-frequency ones.