The short answer is that sampling models aren't not always one or the other: it depends on what you are sampling and the context of the process.
The definition of a random sample in many statistical contexts is basically that you have N observations of a single random variable (X1,X2,...,XN) where they are all independent and also identically distributed.
When you look at the assumptions of with with replacement, it means that all observations have the same distribution so the probability distributions don't change between observations.
Although this is what we typically expect in a random sample, it's not the only way it has to be: for example we can have say an urn with n balls and if we don't replace the ball then this will be sampling without replacement.
All in all, it depends on what the process you are describing is and what kind of distribution it corresponds to.