Rewards will not be synced between devices.
Just passwords, browser history, cookies, settings etc.
The way to “sync” rewards is done by verify with a custodian. Uphold or Gemini currently.
If you don’t verify, the brave browser wallet will be gone if the data is lost. Since it is saved locally on hard drive if the computer.
The wallet you refer to is not where bat get send to… It is completely separated from brave rewards.