"He is putting on a necklace."
Translation:הוא עונד שרשרת.
Some pieces of clothing have their own "wearing verb" in Hebrew, like shoes (נעליים- לנעול) or socks (גרביים- לגרוב). In this case it's about a necklace (שרשרת) and its corresponding verb - לענוד.
This is proper Hebrew, however, a lot a fluent speakers simply use שם ("put on") like you mentioned with all pieces of clothing , I do it too sometimes I have to admit :D
In my opinion just learn it the right way and use the correct words for each piece of clothing, you'll probably see more of these in future lessons.