Do you have any ideas for a password safe that stores its data locally (an encrypted cloud drive is available for synchronizing), offers clients for Linux, Win and Android and has some amenities like filling in passwords in browsers?
My family needs to learn password safety, and I want to make it easy for them.
As other's have recommended: Yes, KeePass.
On desktop i recommend KeepassXC and the plugins for your favorite webbrowser.
And on mobile whatever app works best for you, Strongbox on iOS and KeepassDX on Android are doing pretty great and the integration is also working well.
On mobile you can also save the database password secured with your favorite screen unlocking feature and have webdav, ftp or others available for sync.
On android you could also use nextcloud sync or syncthing.