Does this phrase really exist – уходить по-английский?

Yes, it’s pretty real. Literally means to take English leave, idiomatically - to take French leave (without saying goodbye). Although both idioms are different in English and Russian, they mean the same

It's an actual and common phrase. Уйти по-английски means to leave without saying goodbye.

Same in French: "Filer a l'anglaise" or to leave the English way

And in Central America, the Spanish phrase for leaving without telling anyone or ignoring someone who is calling you is, "Hacerse el ruso", which means "To become like a Russian" :)