Two different Lie groups can have the same Lie algebra

The most important example is perhaps $SO(3)$ and $SU(2)$, both of which have the same Lie algebra, but are not isomorphic.

This simply connected is called the universal covering group.

E.g. in the case of $SO(3)$ and $SU(2)$, $SU(2)$ is simply connected, but $SO(3)$ is not.

The unique group referred to at: every Lie algebra has a unique single corresponding simply connected Lie group.