How can I unlock everything at once and with Touch ID?

@chmars2022

If your device supports the feature and you are using the needed pieces, yes, it should work. Those additional details will help us determine all of that.

An obvious differences is that in Safari, all vaults (local and in the cloud) are visible

Sounds as if you are using a standalone vault and a membership. The extension in Safari would support this configuration. 1Password in the browser does not support this configuration. * See below

while in Chrome, only the vaults in the cloud are visible.

Sounds as if you are using 1Password in the browser. This is (extension) a membership vaults only extension.

Since unlocking the local app is based on the local vault.

Again this is because of the standalone vault. We suggest moving all of your data into your membership and removing the standalone vault (Primary). Once this is complete you'll unlock using the password you set for 1Password.com.

Please reach out to my colleagues and we'll help walk you through the process.