A browser loads websites and mediates what their code can access. Its updates, privacy controls, extension permissions and account-sync settings all matter. The services you visit also receive data directly from your interactions.
A browser’s ownership or marketing message is a reason to examine its design and policy, not proof of what a particular session disclosed. Define the concern first: cross-site matching, local history, account activity or a receiving service’s handling of your query. Different controls address different routes.