$1+\epsilon$ approximation for inapproximable problems
The first sentence says there is no approximation algorithm with $2-\epsilon$ for k-center, which means $k$ is part of an input, because there is no limitation on $k$, so $k$ can be arbitrary large (as $n$). This is common mistake in the context of approximation algorithms and parametrized complexity, this doesn't mean the algorithm runs in polynomial time, this says if you "fix" some parameters (e.g here k) you can have polynomial time algorithm, exactly like what you mentioned, mentioned algorithm is polynomial in $n$ but not in $k$, and when $k$ is not fix and is part of input, the algorithm is not polynomial. You can read more detail on fix parameter tractability in reference question.