Can I use the same user database for both feed and chat?

The Chat and Feeds APIs are different products and therefore they do not share any data so you would need to create the user twice. If you're planning on integrating both, sometimes it makes the most sense to maintain the source of truth on your end so that it's easier to pull certain data.

Users in feeds are different than in chat. For feeds, they are more like extra metadata used to enrich activities. This means that they aren't visible from the dashboard like you would in chat but you can query the API for them.