Yes, the main idea of active listening is to echo back the content, emotions, intent, or identity of the person that they're revealing through the conversation. It's definitely not to try and solve their problems for them... although often times it can help them solve their own problems by giving their thoughts and emotions clarity.

A key distinction I would make here in terms of language is that you're not trying to be sympathetic when you are active listening. Sympathy is showing that yes, you do indeed feel sorry for them/their circumstances.

Rather, I'd say that truly great active listening is about empathy. That is, showing that you can understand and feel their emotions as they do.