Do isotopes that have the same atomic number have the same atomic mass?
No, unless they're the same isotope!
I'll assume by "atomic mass" you don't mean the "average mass of the naturally occurring isotopes", but rather the individual mass of the isotope itself.
If multiple isotopes have the same atomic number, then they are isotopes of the same element.
However, isotopes of the same element can have different numbers of neutrons. For atomic masses, neutrons do not have negligible mass (like the electron), so this therefore indicates different isotopes of the same element have different masses.