Say a random variable $X$ returns one of the values $1, 2, ..., 6$.
Conceptually, does it make sense to speak of a population, where each individual element is just $1$?
In other words, does the population strictly have to cover all possible values for $X$?
The range of a real-valued random variable $X$ (which is a real-valued function) is in principle defined over all $\mathbb R$ (if the random variable is discrete, its range is not $\mathbb R$, but the same approach applies).
But what links the "population" under study and $X$ is the probability distribution, which is a function that describes some characteristics of the range of $X$.
When we therefore define that the probability (density or mass) function, takes the value $0$ for some interval / or set of the range of $X$, this signifies that the population, an aspect of which $X$ models and maps to numeric values, does not take these values for which the density/mass function is zero.
(side-note: strictly speaking "has probability zero of taking these values" -which even more strictly speaking does not make their appearance impossible, only improbable - but I challenge any measure-theorist to come up with an intuitive and practically useful exposition of the difference. It is indispensable for the internal consistency and theoretical validity of the mathematical system, but that's all. But see also @whuber's comment regarding a different and more useful aspect of the distinction "improbable/impossible". End of side-note).
But for every interval/value for which we define a strictly positive density/mass function, it follows that these are values in the range of $X$ that correspond (through the mapping rule that the function $X$ is), to states that members of the population may found themselves into (or so we assume).
Given this general approach, I suspect that when you say that $X \in \{1,..,6\}$, you imply that to each of these six values, strictly positive probability mass is allocated. So the population viewed as a whole, does take these six values.
But maybe you may be asking something a little different : "Does this mean that each member of the population separately can possibly take all of these six values"?
Hmm, what this would imply: Assume that there are only six human-eye colors in the world, and no people have different eye-color per eye. We map each eye-color to one of the six numerical values, and we want to find the proportion of each eye-color in the population. How many eye-colors characterize a single human? Obviously, just one, i.e. conceptually, each human has one eye-color. But to the degree that we do not possess any other relevant information (this is the unconditional probability), is it possible to have any one of them?
The answer touches on how one defines and uses the concept of "statistical population" - a difficult concept. If we answer "no", then in what sense humans belong to the same population as regards their eye color? If 100 humans here can have only the colors A,B,C, and 100 humans there can have only the colors A,B,D, is it valid to "group" them into "one population"?
In other words: should the concept of statistical population includes, as a defining property, the homogeneity (in the sense of possible values/states) of its members as regards the characteristic we want to study?
And what may be the relation of the above to the concept of "identically / non-identically distributed" random variables? When we say "non-identically distributed random variables", are we referring to a different distribution (e.g. same distribution family-different parameters) or possibly also to different ranges with non-zero probability?
