Does google send my session to the browser on android?

[Windows 10]

Hi! Before starting, I want to clear out that I am not very fluent in english.I only have a small doubt. When I am using my android phone and I enter from the google search engine to a page that is from google itself, such as the search history, does google send my account login or data to the web browser that opens the link (firefox, edge, etc.)? I know this is very specific, but I hope you can help me. Thanks in advance. Have a nice day.

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