Well, that would be Hebrew as old as the hills. The oldest Hebrew inscriptions are 3000 years old, the oldes clay tablets in Akkadian are 4500 years old, Protosemitic began to split around 5800 years ago, even creationists who think Adam spoke Hebrew would calculate only with less than 6000 years.
Context and expectations are everything in a language! אצל has a vague meaning of proximity and/or possession. If you use it for shoes, what might you suggest? It could be "at his house", but it seems more likely, I think, that he carries the shoes with him (e.g. "I can't find Jenny's shoes! Have you seen them?" "It's OK, they are with me"). If talking about sleeping, what might you suggest? Most probably, at his house. "Sleeping with him" more strongly suggests sharing a bed, maybe sex, much more strongly than the Hebrew sentence, so it's correct not to translate it this way.
Not only religious Jews. Those of us who grew up in New York were familiar with phrases like "by him", and although generally spoken by Jews, not all were religious. Yiddish speaking immigrants used it regularly, even when not at all religious, and the first generation children of immigrants sometimes did also.
"By him" does not mean "at his place" in the vast majority of English dialects, so it's not considered a correct translation. For most English speakers "by him" can only mean "next to him." As mentioned in other comments above, your interpretation of "by him" is specific to Jewish Americans (maybe only NYC Jews?) and virtually unknown elsewhere.
Didi, maybe this will be helpful for you, the webpage lists the different prepositions (in, at, with, on) to use with the verb "sleep" and how they are used.
23 May 2019
If you read some of the comments here, you might get a better picture, but basically the word אצל is a bit difficult to translate and it's similar to French chez that is idiomatically translated as at someone's place. Since here אצל is written with the 3 person singular masculine pronominal suffix - אצלו the best translation is at his place*.
Even if this did imply sex, which is quite the conclusion to jump to considering the complete lack of context (this really only tells us she's sleeping in his house), it'd still hardly be inappropriate considering how politely and indirectly it's phrased. Most people aren't offended by the concept of sex, if you don't like it then be a grown-up and ignore it.