Hi, perhaps this can make more explicit the correspondence above:

Let $G$ be a $\ell$-dimensional algebraic torus. Denote by $\Xi(G)\simeq \mathbb Z^\ell$ the *character group* of $G$ consisting of all continuous group homomorphisms $G\to \mathbb C^*$. Take any $\chi \in \Xi(G)$, it defines a one-dimensional complex representation of $G$ with space $\mathbb C_\chi$. We can associate a complex line bundle $L_\chi$ over $\mathbb B_G$ by:

$
L_\chi
:=
(\mathbb E_G \times_G \mathbb C_\chi)
\to
\mathbb B_G.
$

Denote by $c(\chi):= c_1(L_\chi)\in H^2(\mathbb B_G)$ its first Chern class. Let $Sym_\mathbb Z^*(\Xi_G)$ be the symmetric algebra of the group $\Xi(G)$. It is a polynomial ring on $\ell$ generators of degree $1$, and the map $c\colon \chi\mapsto c(\chi)$ extends to a ring isomorphism:

$
c\colon
Sym_\mathbb Z^*(\Xi_G)
\stackrel{\sim}\to
H^*(\mathbb B_G)
$

which doubles degrees. It is called the *characteristic homomorphism*.

Now, at an isolated fixed point under the action with characters $\chi_1,\dots, \chi_n$ one has:

$(\mathbb E_G \times_G X) = L_{\chi_1}\oplus\dots\oplus L_{\chi_n}.$

The equivariant euler class is just the top ordinary Chern class $c_{top} (\mathbb E_G \times_G X)$, and you can conclude by Whitney sum formula.