Final answer:
The tunica intima is the blood vessel layer with consistent width in both veins and arteries. It's the innermost layer, whereas the middle layer (tunica media) and the outer layer (tunica externa/adventitia) vary in thickness between these two types of vessels.
Step-by-step explanation:
The layer in blood vessels that is the same width in both veins and arteries is the tunica intima. This is the innermost layer of the blood vessels and is consistent in thickness in both vessel types. Arteries have a thicker tunica media than veins because they need to handle higher blood pressure. In contrast, the tunica externa or tunica adventitia, which is the outermost layer, is also different in thickness between arteries and veins, with arteries having a thicker version to accommodate the increased pressure.