[privacy sandbox] Fix crashes when getting service in the bubble

Save a reference to the service as an attempt to fix crashes when
getting a service later on. Crashes seem to be caused by browser_ reference not being valid. browser_ is valid when dialog is opened
so it must be either browser being destroyed or browser becoming invalid for unknown reason.

Bug: 1379327, 1380342, 1370708
Change-Id: I5e562d7be6ebdb27b63fbeaf82c7fb9f0262166a
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/4040131
Reviewed-by: Theodore Olsauskas-Warren <sauski@google.com>
Commit-Queue: Olesia Marukhno <olesiamarukhno@google.com>
Cr-Commit-Position: refs/heads/main@{#1074055}