fix(browser): Update session after `Sentry.setUser` calls

[github-actions]

Note

The pull request "fix(browser): Update session after Sentry.setUser calls" was created by @Lms24 but did not reference an issue. Therefore this issue was created for better visibility in external tools like Linear.

• fix(browser): Update session after setting user
• done, cleanup

[linear]

JS-1723