The cause of wrinkles in any material, including shirts is a result of hear and water. Like shrinking, the bonds are the problem and with this problem, the bonds are broken and this moves them into different positions. You definitely do not want to be going out in your matching shirt with it wrinkled
An iron removes wrinkles in the same way, by heating up the fibers, new bonds will form that are no longer wrinkled.
Water is also a cause of wrinkles. The fibers have the same molecules as water that hold them together. Like heat, the water penetrates the fibers and create different bonds and form wrinkles.
To try to prevent wrinkles from happening don’t wear clothes that are freshly ironed or out of the dryer or damp as fabric crease easiest when they are warm or damp. Ensure they are dry and cooled down before putting them on.
There are anti-wrinkle sprays available and while they will not completely put a stop to wrinkles but will certainly help in the quest for wrinkle free clothing and you can be safe knowing that when you buy shirts they will last for a long time.
Wrinkles will look pretty terrible when you both go out in your coordinated shirts so have them washed and ironed in the above way and you can both spend the day looking awesome.