I start to think that the problem is not due to Vim in the first place. See this.
Basically, if I send Vim to background via Ctrl-Z, then when I hit fgEnter Vim goes to foreground-and-quickly-back-to-background; if I again press fgEnter, then it successfully goes to forground. And the issue repeats over and over, as you can see in the following GIF:
It first happened today, and I've not been playing around with
.vimrc, nor with
.bashrc, nor other configuration files in general.
Honestly, as far as I know, it could be a Vim thing or a Bash thing.
Plus, I can't give further details for the simple reason that when I open a new session of Vim the problem is absent initally, then it occurs later, but I haven't understood yet what triggers it.