As you wrote, "Like that, if choose different periods, we get different frequencies also." So, I think the problem is the author's saying that a constant signal is periodic "with period \$T\$ for any positive value of \$T\$". The more normal way to understand the period of a constant signal is, that it is \$\infty\$. Otherwise, it can lead to a confusion regarding the corresponding frequency. By duality, frequency could similarly be any value we choose, and hence undefined, if we allow the period to be any positive value of \$T\$.
So, we should define the period of a constant signal to be \$\infty\$, and by duality, it follows that the frequency is unambiguously \$0\$.