It's all because the empty set is the same empty set regardless of "what it's empty of". Yes, the empty set is a "set of singletons" by this reasoning.
Therefore it is false (or anyway, too vague) to include "not singletons" in your proposition: "a relation is a set of ordered pairs, not singletons". The empty set is "a set of singletons", but it also "contains no singletons". So which is that phrase "not singletons" supposed to mean here -- that it doesn't contain any singletons, or that it is not "a set of singletons"? The empty set qualifies by the former condition but not the latter.
You could more precisely say "a relation contains 0 or more ordered pairs and no singletons", and then it's clear why the empty set qualifies. Or you could say "no set of singletons is a relation", which is simply false under the terminology we've agreed, since we've already agreed that the empty set is "a set of singletons" and also is "a set of ordered pairs". You incorrectly excluded the possibility of a set which is both of those things when you extended "a relation is a set of ordered pairs" to "a relation is a set of ordered pairs, not singletons". There was no justification for adding "not singletons".
In general one must be precise when saying "a set of ordered pairs", whether we mean "a set containing zero or more ordered pairs and no element that is not an ordered pair", or "a set containing one or more ordered pairs and no element that is not an ordered pair". By pointing out that the empty set is to be considered a relation, the author makes clear (if it wasn't already) that the former is intended here. If the latter were meant, then you would be justified in saying that "a set of singletons cannot be a set of ordered pairs". But if the latter were meant then the empty set wouldn't be a relation at all, so you wouldn't be worrying about it anyway!
If, via some notion of types, "the empty set of ordered pairs" and "the empty set of singletons" were different objects, then sure, the empty set of ordered pairs would be a relation and the empty set of singletons would not. But that's not the set theory that you're currently working in.