Network share always mounting in Finder, but never for Time Machine (High Sierra 10.13.4)
Problem solved. Here's the reason of all this:


Both Macs were configured with the same iCloud account.
For some reason, the admin user on the Mac Mini had an alias that was the email address of the iCloud account.
The MBP's Finder mounted the network share using that iCloud email address (admin alias). That's probably why it didn't ask any credentials.
But something went wrong with the MBP's Time Machine process, that kept trying logging with the Mini's admin user "real" credentials (not the iCloud alias).


What I did:


On the Mini, in the "Users &amp; groups" preference pane, I right-clicked on the admin user and deleted any aliases found in the advanced options. See this for step-by-step instructions: Prevent File Sharing access with Apple ID
On the MBP, I mounted the network share (using the "real" Mini's admin credentials), then I recreated the Time Machine setup.