As other answers stated, it can be spelled phonetically as "сорян" and is actually adopted "sorry" from English and is used as one.
It can be considered a part of argot or anti-language of younger generation who constantly brush with English in everyday life. As any anti-language, it is a part of informal speech. Consider it a cross-cultural influence. It got a connotation of familiarity or lack of seriousness in the situation. So it can be used only as a lighthearted "sorry": "Sorry, I could not get to your party in time, what did I miss?"
Those who use it, aren't necessary speaking English, they have just got hearing a word or two repeatedly. Common expressions, movie memes (e.g. Arnold's "I'll be back" - "Айл би бэк"), some IT or technical terms, sometimes some trope names that cannot be translated directly, are just transliterated and adjusted to Russian pronunciation.
It's not a new phenomenon, it can be compared to abuse of French language by Russian nobility and those who wanted to be "accepted" as educated person in Napoleonic era. French was butchered due to horrible pronunciation, essentially creating a new "slang".In past, since Peter I Russian language was doing that to Dutch, then to German, then to French, then to Polish and German again. Essentially, to whatever foreign language was playing significant part in culture. A lot of modern military, naval, gastronomic, architectural terms were taken this way.
This can be said about Russian as well:
“The problem with defending the purity of the English language is that
English is about as pure as a cribhouse whore. We don't just borrow
words; on occasion, English has pursued other languages down alleyways
to beat them unconscious and rifle their pockets for new vocabulary.”
― James D. Nicoll