You have to make sure your employer stops withholding Minnesota tax. You no longer work in Minnesota. The employer has to make the change. Ask them if they can withhold Wisconsin tax. From now on your paychecks should have only Wisconsin tax, or no state tax.
A little extra Minnesota withholding this year won't matter. It will get reconciled on the Minnesota part-year resident tax return that you will have to file at the end of the year, so you will get back any excess withholding. But your W-2 is also going to show the income as Minnesota income, even though you are now working in Wisconsin. I don't know how difficult it would be to correct the allocation of income between the two states on your tax returns. So you should get it corrected as soon as possible. Ideally the employer would correct their payroll records for the incorrect check that they already issued, so your W-2 would show the correct amount of income for each state.
(If they can't withhold Wisconsin tax, your W-2 will probably show no Wisconsin income. That's okay, as long as the Minnesota income is correct.)
To be fair, remote work used to be rare. It has only become common in the last few years. A lot of employers are not really oriented to it, and are just figuring out how to handle it. Their payroll and HR systems are probably not designed to accommodate it.
Make sure your employer has your new address in their payroll and HR records. Among other reasons, they will mail your W-2 to the address in their files. There is nothing new about employees moving, but you have to make sure they update all their records.
