Not desktop only, KeePass uses an encrypted file, all you need is a way to decrypt it. You can store it in iCloud or whatever you like to sync files between devices.
iOS has a good open source app KeeForge to open the encryped password files. I use SyncTrain on my phone to connect to my SyncThing network.
I use syncthing for my Linux devices but though it didn’t work on iOS. Will check out SyncTrain, thanks!