KeePass2Android
★★★★★
Nice, clean interface to a KeePass2 database with auto-fill support for both websites and apps.
You can store the database locally, or sync it to a network server - and there are a lot of supported options ranging from your own SSH or WebDav server through NextCloud and up to the major cloud services like DropBox, OneDrive etc. This gives you control over your password storage. Even if you use a cloud service, it can only access the encrypted file.
Syncing is almost seamless, too. I’ll often add a password on my desktop (where I use KeePassXC and its browser extension), then open up my phone and it’s synced already. Or vice versa.