The fundamental group of the Klein bottle is $D_\infty$, the infinite dihedral group (which is $\mathbb Z \rtimes \mathbb Z_2$).
I believed this for some time, and I seem to recall some others having the same confusion.
The group that has been mistaken for $D_\infty$ is in fact $\mathbb Z_2 *\mathbb Z$, though it is usually written with the presentation $x^2y^2=1$ or $aba^{-1}b=1$. They both abelianize to $\mathbb Z\oplus \mathbb Z_2$, which i think was part of my confusion.
A 2-dimensional Lie group is a product of circles and lines, in particular it is abelian.
I don't know if anyone else suffered this one. The mistake is (a) in forgetting that the classification of surfaces doesn't apply since homeomorphic Lie groups are not necessarily isomorphic (so one finds a 2 dimensional Lie group in $GL(2,\mathbb R)$, those whose first column is (1,0) relative to some chosen basis, which isn't abelian but is homeomorphic to $\mathbb R^2$) and (b) that Lie groups aren't necessarily connected, in particular $\mathbb R^2$ cross any finite non-abelian group is non-abelian.