1. Password manager for PC / Laptop: KeePassXC. It's not built into your browser, it's a seperate application. It's totally open source, and trusted by many. It also supports two factor authentication, I use a passphrase and a key file. Supports TOTP. Has a ton of "premium" features, totally free. It's awesome.
2. Syncing application: Google Drive. Sync your KeePass database using Google Drive (or whatever other sync application you want). KeePassXC supports merging databases if there's ever a conflict, as rare as those are. This is secure, because the KeePass database file is encrypted, and Google Drive / Google will never see the unencrypted database.
3. Password manager for phone: KeePass2Android. Not sure what the options are for Apple, but I'm sure they exist. Allows you to open your KeePassXC database from Google Drive.
4. Browser support: KeePassXC-Browser. Allows you to autofill your username / password / TOTP from your KeePassXC application to Chrome / Firefox.
Totally free, secure, convenient, and syncs to all your devices. Also comes with excellent redundancy for your password database so you'll never lose it. I've been using this setup for years flawlessly.