So the distribution is indeed uniform but it fails in being random.
Consider the sequence 1,2,1,2,1,2,1.... When you take elements one at a time, you have a uniform distribution, but the sequence is clearly not random. However, when you take elements two at a time, you have; 12,12,12,12... In this case, the distribution is not uniform, which indicates non-randomness.
The paper's definition implies that you're grouping the sequence elements like this and, for all groupings, you find uniform distributions. This is of course common knowledge to experts, and is made clear in the body of the paper.
Pr[A(prng(seed, n))] - Pr[A(uniform(n))] < 1/negl(n)
Note that (1,2,1,2,...,1,2) is not a "distribution," as you seem to be saying. It's a sequence of values drawn from a distribution.
For a fixed length, is it possible for the sequence (1,2,1,2,1,2,...,1,2) to be drawn from a uniformly random distribution that contains 1 and 2 as possible values? Yes. How likely is it? It depends on the number of possible values in the distribution and the length of the sequence. The more values besides 1 and 2 and the longer the sequence, the less likely that sequence will have been drawn from a uniformly random distribution.
In fact, as the length of the sequence increases, the probability of any particular sequence approaches 0 but is always greater than 0.
The crux of your confusion appears to be that you think "distinguishable" means "a sequence that couldn't possibly have been drawn from a particular distribution," which is not the case.
The notion of "distinguishible" here refers to an experiment where an algorithm can receive one of two possible inputs, with equal probability: a sequence of uniformly sampled numbers, or a sequence generated by a PRNG. If the PRNG is secure, the algorithm should output "1" when it receives the uniform sequence with (very nearly i.e. different by a negligible amount) equal probability as when it receives the PRNG's sequence. So, for example, if you test if even indexed elements are 1 and odd indexed elements are 2, and output 1 if both tests pass, then you will almost never output 1 for a uniform sequence; on the other hand, if the PRNG always and very frequently outputs 1,2,1,2,1,2... then you will always or very frequently output 1 when you get the PRNG's sequence as input, and thus your probability of outputting a 1 is greater in that case.