A Binomial distribution is always over counts of events (where each event has one of two possible outcomes). E.g. the number of heads in a sequence of $n=100$ tosses of an unfair coin with $p=0.2$ has a binomial distribution $B(100,0.2)$.
Your random variable $X$ is the outcome of a single event. If this were an event with only two possible outcomes, then technically you could say this follows a Binomial distribution with $n=1$, but in this special case the Binomial is equal to the Bernoulli distribution.
Since your event (the number recorded on the $i$-th draw) has multiple (namely $N$) possible outcomes, we need a generalization of the Bernoulli distribution. The most general case would be the categorical distribution, which allows for different outcomes to have different possibilities. However, the way your problem is framed means that all outcomes are equally probably, so in that case the categorical distribution reduces to a discrete uniform.