Desktop: KeePassXC

KeePassXC is a cross-platform desktop app that runs on Linux, Windows, and macOS.

The official KeePass software only targets Windows and runs on Linux via Mono, but looks non-native and clunky. KeePassXC uses the Qt GUI framework and looks nice on all platforms, and is compatible with the same KeePass database files.
Mobile: KeePass2Android Offline

For my Android phone, I'm using Keepass2Android Offline because I don't need cloud sync services, nor do I really want to use them. There is also Keepass2Android that supports cloud services, if you want to sync your database via Dropbox or Google Drive or some other services.
Syncing

To sync my password database between devices, I set up a Syncthing instance on my web server (as the centralized host) and my various devices, including my phone. The password DB itself is managed in a Git repository, so after I make changes, I commit it to git to ensure that there is an 'out-of-band' good copy of the database that doesn't get automatically pushed around between devices. Just in case something happens.