Q. Are phrases like “The more the merrier” and “The bigger the better” sentences, despite not having a verb?
A. In CMOS 6.50, on commas with “the more,” “the less,” and the like, you’ll find one of those as an example that illustrates when a comma is unnecessary: “The more the merrier.”
As you suggest, that’s a phrase. But when it’s standing alone like that (with an initial capital and a terminal period), it functions as a type of sentence—one that’s elliptical, meaning that one or more words have been left unstated but will be understood by readers, sometimes with the help of context. The adjective elliptical comes from the Greek verb elleipein, “to leave out,” as does the noun ellipsis, which can refer either to an omission of words or to the series of dots (. . .) that can mark such an omission.
If you supply the missing words in your two examples (as we’ve done below, choosing a generic reading for both), it will be clearer that they work as complete sentences, though with some inversion:
The more there is, the merrier it will be. (= It will be merrier to the degree that there is more.)
The bigger it is, the better it will be. (= It will be better to the degree that it is bigger.)
Note the commas, which become necessary when words like the ones underlined above are expressed rather than implied: “The more people we invite to our grammar party, the merrier it will be.”
For the related phenomenon of elliptical sentences like Why? and Yes, see CMOS 5.101. For ellipses (the dots), see 12.59–69.