Why “the Aye is 51, the Nay is 48,” not “Ayes are 51, Nays are 48”?

The syntax you heard could be construed as the result of an elision, for example of "vote" or "vote count":

The aye [vote count] is 51; the nay [vote count] is 48.

That elision, in turn, may have been prompted by the reading of an electronic display. Hypothetically, that is, the Senate Legislative Information System may provide the Secretary of the Senate with a display showing the number of "aye" and "nay" buttons pressed by senators when voting. The Secretary (or clerk) may then read the electronic display out loud. The counts may even be labeled "Aye" and "Nay" on the electronic display, thus reinforcing the elision and so the use of singular syntax.

For background procedural information, see "Roll Call Votes by the U.S. Congress"; see also "Quorum and Vote" in "Procedures of the United States Congress".

More generally, however, underlying linguistic processes condition fluent speaker understanding of singular syntax and the corresponding absence of plurality markers for what would ordinarily be countable nouns. Those processes are described in detail in, for example, the 1980 paper "Nouns and Countability", by Keith Allan. The abstract of his paper provides an adequate summary:

The customary disjunctive marking of lexical entries for English nouns as [± countable] does not match the fact that the majority can be used both countably and uncountably in different NP [noun phrase, ed.] environments: this binary opposition is characteristic not of the nouns, but of the NP's which they head. Nevertheless, nouns do have countability preferences; some enter countable environments more readily than others.

Thus, although I hesitated a bit over the syntax you encountered, I did not, finally, register it as incorrect or nonstandard. The nouns 'aye' and 'nay' appeared in an environment that enabled me to readily construe them as uncountable despite their "countability preferences".