Is the empty set countable?

The empty set is a subset of $\mathbb{N},$ therefore a countable set.

For motivation, the intersection of two countable sets is a countable set, and the intersection of any two countable disjoint sets is the empty set.

In mathematics, as in all other technical fields from physics to furnace repair, there is a technical jargon in which ordinary terms, like “countable”, are used in particular technical ways that only somewhat resemble the ways the same words are used in ordinary speech. It is useful to think of the ordinary meaning of these words, as an aid to understanding them, but that ordinary meaning only gets you so far, and not as far as you need to get.

To understand “countable”, you can think of ordinary counting, but that will not get you all the way to understanding the mathematical meaning of “countable”, because the mathematical meaning is not the same as the ordinary meaning. In mathematics, “countable” has a particular technical meaning that you cannot guess from your ordinary understanding of this word. To understand the mathematical meaning, you must look up the mathematical definition.

The mathematical meaning of “countable” is:

A set $S$ is countable if there exists a subset of the natural numbers, say $T$, and a one-to-one correspondence $f$ between $S$ and $T$.

To understand this definition, you should look at how it corresponds to the ordinary notion of “countable”, and also how it does not. For example, the definition implies that the set of even numbers is “countable” in this technical sense, whereas the ordinary meaning of “countable” probably does not apply to the set of all even numbers, which has no beginning and no end. Similarly, the definition implies that the empty set is “countable” in this technical sense, although you have observed that to call the empty set “countable” in the ordinary sense is strange. But this just shows that the two senses are different, and you cannot use the ordinary sense as a replacement for the technical sense.