This is not an unusual surprise- many people make an early withdrawal without fully understanding the tax consequences.
A hardship withdrawal does not relate to taxes. When you have a hardship withdrawal from a 401(k) or 403(b), it only means that they let you get your hands on some money. Having a qualifying hardship in the eyes of your 403(b) custodian does not mean you are exempt from the early distribution penalty.
Holding out 10% is rarely enough, because you have to cover both your income tax liability and the penalty. So, while you may have withheld enough to cover the penalty, you didn't withhold any extra to cover regular tax liability.
You don't pay taxes when they withhold. They are withholding in anticipation of what you would owe on your tax return, which is a combination of all of your income. This is essentially the same as when you work- they withhold from your pay, and hopefully they withhold enough. Your tax liability is not calculated until you prepare your tax return, as we don't have a flat tax system.
"Hardship" is not an exception to the penalty, but there are some exceptions laid out in this IRS publication:
https://www.irs.gov/taxtopics/tc558.html
Many of them are just not common, so I will list what I believe to be the 2 most common below:
Distributions made to you after you separated from service with your employer if the separation occurred in or after the year you reached age 55, or distributions made from a qualified governmental defined benefit plan if you were a qualified public safety employee (state or local government) who separated from service in or after the year you reached age 50.
Distributions made because you're totally and permanently disabled.