Feature Request: Expand syncing feature to also sync metadata of other type of accounts

Nobody has claimed this yet.

⚠️ ⚠️ ⚠️ ⚠️ **Update: This ticket was originally opened as a bug but the feature itself don't support other accounts that are not tied to a SRP inside MetaMask. This ticket has been updated to be a "feature request" instead of a bug**

### Describe the bug

I think something with sync breaks the select recipient window. I installed MM on new pc and named everything properly this time and it looked fine but when i tried to send something i noticed that my addresses were showing default name. Tried syncing unsyncing but nothing changed. Some of them are properly named from old pc but not everything. In the final review windows the account name is shown properly. Using latest MM on edge browser. On firefox they show fine but its completely different UI there. Here are screenshots to better describe the issue:

Account names: https://i.imgur.com/6tc85dB.png
Select recipient to send: https://i.imgur.com/BckX8Wp.png https://i.imgur.com/7uh72mM.png
Review part shows correct account name: https://i.imgur.com/uYQLC5O.png

I only named few wallets on my old pc and the ones that sync'ed correctly were the ones imported with seed phrase and contacts. Ledger and private key imports seem to default to default name. Not sure how reproducible this bug is but its certainly annoying because i have no way to fix this.

While typing all this i suddenly remembered i forgot to turn off sync on mobile and this turned out to be the solution. Seems like recipient tab on pc (edge but used chrome store extension) is only syncing with mobile account names which explains the wrong ledger names because i never imported them there and used private key instead of seedphrase on some accounts so they dont match with pc. I guess i can fix it by matching all seed phrases and private keys but its certainly annoying bug that shoudn't happen. I tried disabling all sync and renaming but those names are stuck with mobile names. The backup should have been deleted according to the description of the option.

EDIT: Just tried syncing all accounts by naming them in mobile and seems like all ledger and private key names won't sync properly. Only accounts with seed phrase sync that recipient tab properly.

No files or tests are identified in the issue. Start by reproducing the recipient-tab behavior across Edge or Chrome, Firefox, and mobile with Ledger and private-key accounts, then trace the sync flow for account names. Done means metadata for those account types syncs correctly and the recipient window shows the matching names.

Written by the indexing model from the issue text.