The reason why ${\mathbb C}^*$ is called torus is clear by looking at it real points. It could be a split real torus ${\mathbb R}^*$ or a compact torus $U_1({\mathbb C}) \cong {\mathbb R}/{\mathbb Z}$.

In general, a torus in a real Lie group looks like $({\mathbb R}^*)^n \times {\mathbb R}^m/{\mathbb Z}^m$. The split part $({\mathbb R}^*)^n$ is a bit like $({\mathbb C}^*)^n$. The compact part ${\mathbb R}^m/{\mathbb Z}^m$ is a topological torus.

The reason why ${\mathbb C}^*$ is called torus is clear by looking at it real points. It could be a split real torus ${\mathbb R}_+^*$ or a compact torus $U_1({\mathbb C}) \cong {\mathbb R}/{\mathbb Z}$.

In general, a torus in a real Lie group looks like $({\mathbb R}_+^*)^n \times {\mathbb R}^m/{\mathbb Z}^m$. The split part $({\mathbb R}_+^*)^n$ is a bit like $({\mathbb C}^*)^n$. The compact part ${\mathbb R}^m/{\mathbb Z}^m$ is a topological torus.