Suppose that $X$ takes on $n$ values index by $i$ with probability $p_i$. Let $D_t$ denote the number of distinct values at time $t$. Let $X_s=1$ if at time $s$ we introduce a new value that hasn't been seen before. Then:
$$D_t=X_1+...+X_t.$$ This is essentially the Coupon Collector Problem with generalized probabilities for each coupon. In particular the distribution of $X_i$ is related to the waiting time $W_k$ to collect $k$ distinct events: $P(D_t=s)=P(W_s\leq t,W_{s+1}>t)$. So it's not surprising that something like the normal distribution pops up here, if one assumes the correlations between $X_i$ are small, so that the CLT would apply. If the probabilities of each $X_i$ are tiny and the number of objects is extremely large, then you might expect a Poisson distribution. In particular the waiting time for the next object is geometric: your success probability is the sum of probabilities over all objects you have yet to collect.
I believe originally Baum and Billingsly originally studied the distributions of $W_k$ as a function of both $k$ and $n$ for the classical coupon collector problem (with equal probabilities) and found there are at least 4 different limits for $W_k$: degenerate, poisson, gaussian or gumbal, depending on the way $k,n$ scale with eachother. You can find a summary of these on page 3 of this reference. I strongly presume that similar results will hold for $D_t$, but the specifics for generalized probabilities are likely much more complicated to analyze. However, for a finite number of objects, the ultimate limit shapes are most likely the above distributions.