Yes, both versions would be accepted. "...like I was their own son" is more common, but technically, the subjunctive mood should be used "...as if I were their own son." [The subjunctive 'were' is correct, but in the U.S. anyway, many people commonly, and incorrectly, would use the past tense 'was' here.]

Scott gave a good answer. For English exams, the subjunctive is expected because it expresses an unreal situation. This is closely related to a type 2 conditional. [type 2 conditional with unreal condition (if ...)] If I WERE/WAS their son, then they would treat me very well. [unreal situation (as if ...)] They treat me as if I WERE/WAS their son. More information here: http://guidetogrammar.org/grammar/conditional.htm Also, La Liseuse, a very respected italki member, gave a good explanation of the subjunctive here: https://www.italki.com/question/493340 For casual use, "as if I am their son" sounds quite odd, but there may be some regions where it is said.

Sounds fine and is totally understandable, but it doesn't sound native to me.

Your version is completely fine. The book version is fine too. It's just not the only OK way to write this. https://books.google.com/ngrams/graph?content=like+I+am+their%2CI+were+their+own+son&corpus=26&year_end=2019&year_start=1800&smoothing=3&direct_url=t1%3B%2Clike%20I%20am%20their%3B%2Cc0%3B.t1%3B%2CI%20were%20their%20own%20son%3B%2Cc0