What is the need for torsion in the definition of lisse sheaves?

I think you are misinterpreting things slightly. Lemma 2.1 says nothing about abelian groups. Lemma 2.1(i) is about sets, and Lemma 2.1(ii) is about A-modules. For $A = \mathbb Z$, $\mathbb Z$-modules are equivalent to abelian groups and so Lemma 2.1(ii) applies.

It is only 2.1(i) that is used in Remark 2.3(i). That is used to show that a sheaf of groups that is constructible as a sheaf of sets is also constructible as a sheaf of groups - in this case we take $f$ to be the multiplication or inversion map. The last line, that $\mathbb Z_X$ is constructible as a $\mathbb Z$-module, but not as a sheaf of groups, follows immediately from the Definition 2.3, as of course $\mathbb Z$ is constant and of finite presentation, but isn't finite. Nothing about the category of sheaves of $\mathbb Z$-modules or the category of sheaves of groups forces you to do that, because they are equivalent categories.

So why did Grothendieck define constructible sheaves of abelian groups to be finite? I don't know, but I believe it's exactly for all the other reasons one must work with torsion coefficients. Remember that if Grothendieck wants to talk about constructible sheaves of groups without the torsion condition, he can just work with constructible sheaves of $\mathbb Z$-modules.