client/web: restrict full management client behind browser sessions

Adds `getTailscaleBrowserSession` to pull the user's session out of
api requests, and `serveTailscaleAuth` to provide the "/api/auth"
endpoint for browser to request auth status and new sessions.

Updates tailscale/corp#14335

Signed-off-by: Sonia Appasamy <sonia@tailscale.com>
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