$\pi_1$-equivalence of CW-complexes

Definition. We call $\pi_1$-equivalence the closure of a binary relation between topological spaces "there is a continuous mapping inducing an isomorphism of fundamental groups" to an equivalence relation.

Questions:

1. Are there two non-$\pi_1$-equivalent CW-complexes with isomorphic fundamental groups? (here Achim Krause gave an example of two CW-complexes with isomorphic fundamental groups between which there is no continuous mapping inducing isomorphism, but are there any invariants that could impede the chain of such mappings?).
2. If so, what are the $\pi_1$-equivalence invariants of CW-complexes?

It is clear that a CW-complex is $\pi_1$-equivalent to its 2-skeleton, so we can restrict ourselves to two-dimensional CW-complexes.

For any pair of 2-dimensional (connected) CW-complexes $X$ and $Y$, if $\pi_1(X)$ is isomorphic to $\pi_1(Y)$ then there exist maps going each way $X \mapsto Y$ and $Y \mapsto X$ that induce isomorphisms on $\pi_1$. Using what you have already written that the 2-skeleton is $\pi_1$-equivalent to the whole complex, it follows that the only invariant of $\pi_1$-equivalence is the fundamental group itself.

For the proof, first pick $0$-cells $p \in X$ and $q \in Y$ and an isomorphism $\phi : \pi_1(X,p) \to \pi_1(Y,q)$. Now define $f : X \to Y$ as follows. Pick a maximal subtree $T$ in the $1$-skeleton $X^{(1)}$, hence $T$ contains each $0$-cell. Define the restriction $f \mid T$ to be the constant with value $q$.

Next, orient and enumerate the edges $\{e_i\}_{i \in I}$ of $X^{(1)}-T$. Associated to each $e_i$ is an element $[\gamma_i] \in \pi_1(X,p)$ where $\gamma_i = \alpha_i e_i \beta_i$, $\alpha_i$ is the path in $T$ from $p$ to the initial $0$-cell of $e_i$, $\beta_i$ is the path in $T$ from the terminal $0$-cell of $e_i$ to $p$. So far the map is already defined on $\alpha_i$ and on $\beta_i$ to map those paths to the single point $q$, because $f$ is already defined on $T$ so as to take all of $T$ to $q$. Therefore $f$ is already defined on the edge $e_i$ so as to map the endpoints of $e_i$ to the single point $q$. Now extend $f$ over the rest of the edge $e_i$, defining $f \mid e_i$ to be any closed path in $Y$ based at $p$ which represents $\phi[\gamma_i]$.

It follows that, so far, $f$ has been defined on the closed path $\gamma_i = \alpha_i * e_i * \beta_i$ based at $p$ so as to take $[\gamma_i]$ to a closed path $f(\gamma_i)$, based at $q$, representing $\phi[\gamma_i]$.

Finally, for any $2$-cell $d$, its attaching map $\chi_d : S^1 \to X^1$ is homotopic to some path of the form $\gamma_{i_1} \cdots \gamma_{i_K}$, and then since $[\gamma_{i_1} \cdots \gamma_{i_K}]\in \pi_1(X,p)$ is the identity it follows that $\phi[\gamma_{i_1} \cdots \gamma_{i_K}] \in \pi_1(Y,q)$ is the identity and so $f(\gamma_{i_1} \cdots \gamma_{i_K})$ is homotopic to a constant in $Y$. From this it follows that $f \circ \chi_d$ is homotopic to a constant in $Y$. Using this homotopy we can therefore extend $f$ continuously over $d$.