Yes, there is such a thing as “Mormon underwear” although Mormons refer to them as “garments” or “temple garments”. Only Mormons who have “gone through” a Mormon temple (“temples” are different than “churches”), meaning they’ve participated in certain religious ordinances, wear religious clothing as underwear which is called “garments”. The clothes serve as a reminder of the promises Mormons make in the temples and Mormons believe wearing the garments afford them a certain level of physical and spiritual protection. Mormons consider the clothing sacred and therefore don’t like to show them off or let them be seen for fear of someone making fun of what something that is very meaningful to them.
As a practical matter, the garments are a very effective reminder of the commitments made in the temple. Every day I get dressed or undressed I’m reminded of who I am and the promises I’ve made as I put on or take off those underclothes. One might think wearing these underwear an inconvenience, but that’s a bit of the point. If they didn’t get in the way, they wouldn’t serve as a reminder. They are inconvenient only in the same way putting a pile of papers for work in front of the door so I don’t forget them makes it inconvenient to get out the door and get to work. I want to remember the commitments I made, and the temple garments assist me in that objective.
Of course this may all sound ridiculous to someone who isn’t familiar with the practice, but many other religions also have dress codes wherein the members of that faith wear certain clothing or objects as symbols of their beliefs. Mormons can be grateful that their underwear hasn’t received the attention of the Muslim burqa. What attention Mormons do receive for their garments/underwear would appear to have more to do with the titillating fact that talking about it allows otherwise mature people to use the word “underwear” in conversation.