It indeed works for chats as well. If you have a busy forum, you can configure it so that it only works for chats. This helps keep the noise on your phone at a reasonable level.
Be aware though, the response time is not at the level of a native app. It typically takes 2-10 minutes for notifications to come through if you aren’t actively using the PWA. If you need instant notifications, then the expensive and rather high maintenance app packaging is probably the way to go.