It is said that Lorentz group $O(1,3)$ is a six-dimensional noncompact non-abelian real Lie group that is not connected. The four connected components are not simply connected, but rather doubly connected. https://en.wikipedia.org/wiki/Lorentz_group#Basic_properties
What is the definition of the doubly connected? Does it mean that $\pi_2(O(1,3))=0$?
In this context, "doubly connected" means that $\pi_1$ is a group of order $2$. $SO^{+}(1, 3)$ has universal cover the indefinite spin group $\text{Spin}(1, 3)$, which turns out to be isomorphic to $SL_2(\mathbb{C})$.
This isn't great terminology because, as you say, it can be misread as meaning $2$-connected, which means that (the space is path-connected and) both $\pi_1$ and $\pi_2$ vanish. Some people also use "$n$-tuply connected" to mean that there are $n$ path components; I would avoid that terminology too.