LTM

This depends on the persistence you have configured for the Virtual Server.

For source address then the persistence is done on the Client IP address, as seen by the F5, so all traffic from the client would be sent to the same pool member regardless of how many browsers they had open.

For any cookie persistence it would be based on the browser, so if a user had two different/independent browsers open each would have it own cookies and so persistence would be per browser.

Think about what did of data the F5 would look at to create/lookup persistence records, e.g. if something is sent by the browser (cookies and headers) then persistence will be per browser.