This is a bit of a false dichotomy: both blue and red were very common colors across Europe in the Middle Ages and beyond, and both could be used to show status as well. There was no special distinction for all of blue as royal, and red was not disdained by royalty. Quite the opposite! The important and expensive thing was to get a very deep color. (Perhaps you are thinking of purple? We do see early modern - and Roman - sumptuary legislation that restricted purple to royalty.)

Blue dye was widely available from woad, and red from madder - both grew across Europe and formed the backbone of the local dye industry. However, neither was perfect. Madder tended to produce a kind of orangey or pinkish tone, and woad was generally a little greenish. European monarchs, nobility, and rich people in general wanted better - something more like the deep tones found in the illuminations painted using mineral-based colors.

In contrast, true indigo contains more dyestuff and produced a much more striking blue, especially in linen and cotton, the latter of which was an incredibly expensive import until the 18th century, when it became a moderately expensive import and then a fairly cheap fabric. That said, woad could achieve the same results in wool and silk - it's just that more dyestuff would have to be used by the dyer. (So either way you're paying more.) A true red was achieved first through the use of kermes, a tiny insect native to the Mediterranean, which could be dried and ground to produce a striking and vibrant shade, as seen on Roger II of Sicily's coronation mantle; a genetically unrelated but similar New World bug called cochineal became a very common source for this shade as well, following the Spanish conquest.

And this is a digression, but I think it helps to illustrate my point: another quite important color was black. There's a perception that the current use of it as a signifier of cool is a very modern thing and that people used to wear it solely for mourning or general dourness (e.g. Puritans), but it was in fact a great status symbol in the fifteenth, sixteenth, and seventeenth centuries, and you'll find it liberally worn in portraits of the royalty/nobility/prosperous merchantry in order to show off the expense. It was achieved in a few different ways, but the best was through overdyeing - putting the cloth into multiple dye vats of different colors - and so black's use as a status symbol was more about paying for the skill needed to get a rich color than paying for an expensive dyestuff.