Yes, a User Specific array would work, but you would need a Helper Table to expand that array into a list so that they could select from a list of tags, as well as manage the list.
I’d recommend using a User Specific Multiple Files column in the User Profile row to hold the array of tags. To add a tag to the array, use a Make Array column with a Set Column values action. And to remove a tag from the array, use a Remove Element column with a Set Column Values action.
For the Helper Table, you could utilise the Miracle Method.