Is this an officially licensed Fourth Wing shirt?

No license is stated. It is a Temu listing using the Basgiath War College name from Rebecca Yarros's Empyrean series, so treat it as unofficial fan merch.

What sizes does it come in?

The listing is a women's cut, but the size range is not given here. Check the Temu size chart and measure a shirt you own before ordering.

How should I wash it?

It is listed as washable. Turn it inside out and wash cold to protect the print, and skip the dryer to reduce fading and cracking.

What is the design on the shirt?

It carries a Basgiath War College Dragon Riders graphic, a reference to the war college in the Fourth Wing books. The exact artwork is shown in the listing photos.

Is it worth buying over a plain gray tee?

You are paying for the Basgiath graphic, not the fabric. A plain tee will likely be thicker and last longer, but it will not carry the Fourth Wing reference.