Basically, there has to be some (hidden) way for Boris to communicate what the secret object is to Alex, probably by either indicating when the question to which he should answer "yes" is coming up, or by signalling how many times to say "no" before answering "yes".

My guess is that Alex knows to answer "yes" to the next question when

Boris asks about something that shares the same first letter as the first guess.

The guesses that precede the correct answer in the examples are "night sky," "shadow," and "crow". In each example,

These guesses are preceded by "napkin," "spoon," and "cigar," respectively.

Basically, there has to be some (hidden) way for Boris to communicate what the secret object is to Alex, probably by either indicating when the question to which he should answer "yes" is coming up, or by signalling how many times to say "no" before answering "yes".

My guess is that Alex knows to answer "yes" to the next question when

Boris asks about something that shares the same first letter as an earlier guess.

The guesses that precede the correct answer in the examples are "night sky," "shadow," and "crow". In each example,

These guesses are preceded by "napkin," "spoon," and "cigar," respectively.

This also fits with the OP's example in a comment, as

The second-to-last guess, "bed," is preceded by a guess of "bottle".