If you use one browser to “login to services” you can use that browser to login to your sensitive services too. If a website you visit does not include any third party stuff (fonts, ads, analytics, …) and the website owner does not collect and share user data with a third party. No third party will know if you ever visited that website.*
If the website is especially privacy respecting it will even set the “no-referrer“ policy for links. So when you follow a link on that website, the website linked to wont know where you are coming from.
(* Yes your DNS provider and your Internet provider or VPN provider will know, but that can’t be influenced by the choice of browser or how you compartmentalize your browsers.)