I haven't seen anything to detect this as far as a tag on Ignition. Even when the drive is full the database connection would still be alive and well. Remember, the database can never get full, it's the drive that runs out of space. You may need to have something on the database side that checks for drive space.
The system tags are what we have to give you. If the disk is full, it's up to the DB to report that to clients (like Ignition) by failing incoming connections. If it doesn't do that, there's not a lot you can do inside of Ignition.
Maybe you can come up with some kind of query you can run periodically that reports from inside the DB, but it'll be specific to whatever your flavor of DB is.
I saw your connection was named MySQL, so that query should give you every table size in a MySQL/MariaDB schema. If you are pointing your tag historian to your MySQL instance, it should include it.
I know some of these are partitions, event data, alarm data, etc. Which of these tables is the "active" tag history data table? (the one that I would need to check to see if the disk is full or not).
Would it be better to enable/setup data pruning on your history provider? You should be able to estimate the storage needed from your history Prune Age.
Technically, the answer is "whichever table is identified by a row in sqlth_partitions where sqlth_partitions.start_time < CURRENT_TIMESTAMP() <= sqlth_partitions.end_time".
Practically, the answer is "whichever table has the right drvid (the _1_ in your screenshot) and the higher calendar index" until/unless you change your partitioning scheme.