While studying complex Lie groups theory, and more generally complex geometry, I've found two different objects which are called "complex tori".
Consider the multiplicative group $\mathbb C^*$, of course this is a non-compact complex Lie group. The direct product of $n$ copies of this is of course a complex Lie group which is referred to as "complex torus".
Let $\{v_1, \dots, v_{2n}\}$ be a set of $\mathbb R$-linearly independent vectors in $\mathbb C^n$, then they generate a lattice $\Lambda = \bigoplus_j v_j \mathbb Z$ which is a discrete subgroup of $\mathbb C^n$. As its action by translation is a covering action we can induce a complex Lie group structure on the quotient manifold $\mathbb C^n/\Lambda$. This is a compact complex Lie group which is also referred to as "complex torus".
Now my question is: if these constructions produce objects which are not even homeomorphic, why do they have the same name? Is there a deep reason for that? To me, it's clear that it's reasonable to call the second ones "complex tori" as their underlying smooth manifold is a torus in the real sense. The only guess I've been able to make for the first construction is that $(\mathbb C^*)^n$ retracts to $U(1)^n$, which is homeomorphic to a torus in the real sense.
Thank you for any insight on this topic
Your reasoning is basically correct. They have the same name because they're both "complex tori" in a sense. It depends on your setting on which terminology one uses/prefers for these things, and here the word "complex" is used for two different things.
In my setting, any space which is homotopy equivalent to an $n$-dimensional torus is a torus; we wouldn't usually consider these objects as sub-objects within an ambient space, and we'd ignore the "complex" bit. If we refer to the objects in 1. as complex tori, we are trying to emphasise certain algebraic structure. If we refer to those objects in 2. as complex tori, it's because we want to emphasise that they're quotients of complex space.