People still cared about race, religion and ethnicity but if you were a big money maker and a dangerous killer as Jimmy Burke was, people that were formally made in the Mafia were going to respect you.

Everyone in NY knew who Burke was and he got respect not only because of his association with the Luccheses but in his own right.

"When the snows fall and the white winds blow, the lone wolf dies but the pack survives."
Winter is Coming

Now this is the Law of the Jungle—as old and as true as the sky; And the wolf that shall keep it may prosper, but the wolf that shall break it must die.
As the creeper that girdles the tree-trunk, the Law runneth forward and back; For the strength of the Pack is the Wolf, and the strength of the Wolf is the Pack.