I got some help from this thread: Allow to open task details when multi-selecting tasks and also found that once you select the Tags column to be visible, after you multi-select, you can click on the tag column and add a tag like that (the comments I read said that you can tab through the columns and do it like that also).
I don’t know if I would have found it without the help of the forum here. It’s a completely new way of doing it, and no tooltips appeared to show me that - they only showed up on the multi-select popup (and keep showing up even after I click ‘Got it’).
Maybe Asana could also add ‘Add tag’ to the multiselect popup (and not require the column to be visible!), that’d be helpful.