A port does not have a characteristic impedance. It has an input impedance, or a port impedance, or just an impedance.
The characteristic impedance of a line is the ratio of V to I that will produce a travelling wave on the line. It doesn't tell us what the actual ratio of V to I at any point on the line until we also know the magnitudes and phases of the travelling waves in both directions are.
The input impedance of a port tells use the actual ratio of V to I at the port terminals due to an AC excitation.
We also sometimes talk about the characteristic impedance of a system. This just means we'll design our system with a default characteristic impedance for the transmission lines in the system. (We might also choose to use different characteristic impedances for different parts of our system)
Once we have defined a characteristic impedance for our system, we can determine what the reflection coefficient is for a port with a given port impedance if it is used in that system. If we actually used lines with different characteristic impedance than the system characteristic impedance, we'd have to do some additional math to calculate the actual reflection coefficient where the line connected to a port, since both the line and the port will have non-zero reflection coefficients in that system.