You will be able assign the gender of your avatar in Fire Emblem: Three Houses, but that will be extent of their customization, Nintendo clarified in a statement to USG today.
Nintendo's statement came in the wake of Eric's Fire Emblem: Three Houses preview yesterday, where he briefly alluded to a "customizable avatar." A Nintendo representative responded, "Although you can choose Byleth's gender and, of course, assign character skill sets, the actual avatar is not customizable in terms of look."
This decision by Intelligent Systems is a little bit of a departure for the series. In Fire Emblem Fates, for instance, you could choose between hairstyles and hair color. In Fire Emblem: Awakening, Robin's appearance could be chosen from a variety of options, from gruff older man to a young woman.
While Nintendo didn't explain why Intelligent Systems opted to drop this option, it's worth noting that this is the franchise's first appearance on Switch. Additional power often adds to the time needed to create new assets, and any appearance options have to be reflected throughout the entire game.
In Fire Emblem: Three Houses, Byleth is a professor at a school called the Officer's Academy. There appears to be some sort of time jump involved, but Intelligent Systems is being cagey about what exactly that entails (though there will apparently be romances, because of course). Eric went hands-on with Three Houses, focusing in particular on the implications of the iconic Weapon Triangle being dropped.
We're just over a month away from the final release, so we'll learn more soon. Fire Emblem: Three Houses will be out July 26 on Nintendo Switch.