A function $f(t)$ is said to be periodic with period $T$ if the following holds for all $t$:
$$ f(t)=f(t+T). $$
As such, the constant function is clearly periodic and every real number is a period. However, it is common to define 'the period' of a function to be the smallest real number that is a period. For example the function:
$$f(t)=\sin 2t $$
is periodic with period $2\pi$, but it is also periodic with period $\pi$. As such whilst you might strictly say "$2\pi$ is a period" if you asked for 'the period', you'd find the answer "$\pi$".
But since every number is the period of the constant function, and there is no smallest positive real number, we arrive at a contradiction and there is thus no single period. This is the source of the tension that "the constant function is periodic, but its period is undefined".
However, this is really a purely terminological issue. In any given case, if you have a theorem relating to period functions (eg maybe you know that in a periodic potential, the solutions to the time-independent Schroedinger equation are governed by Bloch's thoerem) then the constant case is probably a much easier special case that you can solve independently. You will probably find that the constant function does satisfy most theorems about periodic functions but you may need to be careful on occasion.
NB As User SolomonSlow points out, a better terminology is to distinguish between "a period" and "the fundamental period". However it appears that the OP is specifically quoting a text where the word "fundamental" does not appear, which I would say is somewhat common.