Having has multiple meanings. It depends on context. There are sentences where you could replace "having" with "possessing" and sentences where that wouldn't work at all. Context is important even when you're replacing words with only one meaning, because often the available synonyms aren't exact synonyms, and all have subtlety different connotations. Often it can be more eloquent to just say "having" rather than use some fancy word that doesn't quite fit. It's only more eloquent if you actually want the added connotation provided by the synonym.
