Unfortunately, the IRS view on unclaimed funds is that they are taxed according to what they were originally. (Wages should be taxed as wages, if the funds represent a dormant stock account you have to estimate your basis and gains, and so on.) So technically, this is an HSA distribution.
If you contacted your current HSA and told them you had rollover funds, they would probably accept the rollover, as long as it was within 60 days of receiving the money from the unclaimed property fund. However, the IRS might have a problem since the money was withdrawn from the old bank more than 60 days ago, so they might send you a letter asking questions. You can reply with a description of what happened, copies of the letter from the unclaimed fund, copies of the check, and showing that you completed the rollover within 60 days. It might be allowed.
Or, keep the money. Do you have any medical expenses over the past few years (since the time that old account was opened) that you have not submitted reimbursement for? You can apply the money toward old medical expenses, or toward new medical expenses that occur before the end of 2023 (if you got the check in 2023). If you use up the money for qualified medical expenses, then none of it will be taxable.
Or if you have no medical expenses to use up the money, then technically it is a non-qualified HSA distribution subject to income tax and a 20% penalty.
I suspect the state won't send any 1099, or if they do, it will be a generic 1099-MISC. Once you decide whether you will try a rollover or use the money for qualified expenses, then wait and see if you even get a 1099, then next year you can report it in a way that will reflect your treatment of the money now (as taxable or not taxable for one reason or the other).
