I would ask your friend whether he considers this an equation: $$x = x$$ It does have one variable...
Now, I'm sure our learned colleagues can point out some esoteric domain in which the equality of an object to itself is not a foregone conclusion. But for most practical purposes, it is an useless equation, as it does not really tell us anything we didn't already know (before you say that neither does 3 + 2 = 5, it does tell us that the base of numeration is not 2, 3, 4 or 5).
However, by the Merriam-Webster definition, $x = x$ is not an equation, because it involves only one expression ($x$).
mathematics : a statement that two expressions are equal (such as 8 + 3 = 11 or 2x – 3 = 7)
The big difference between $3 + x = 5$ and $3 + 2 = 5$ is that, although they are both equations by the dictionary definition, only one of them needs to be solved, the other one already is.