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 $\operatorname{Sym}_\mathbb Z^*(\Xi_G)$ 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 \operatorname{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:
$(\mathbb E_G \times_G T_X) = L_{\chi_1}\oplus\dots\oplus L_{\chi_n}.$
The equivariant euler class is just the top ordinary Chern class $c_{\operatorname{top}} (\mathbb E_G \times_G T_X)$, and you can conclude by Whitney sum formula.
