Caveat emptor: the European Middle Ages was a big place, and I can't pretend to speak for all of it. My comments are largely confined to western Europe.

What I can say is that in most cases, western European fighting men had the responsibility of equipping themselves for battle. With few exceptions (more on that in a minute), they were not provided with arms and armor by anyone or anything, certainly not by a regular military establishment. Outside of Byzantium, standing armies did not exist in the period, full stop. The very first state-supported full-time professional soldiers only appeared in the late 15th century. Rather, medieval armies were built from some combination of aristocrats and their personal followers, mercenaries, urban militias, and (sometimes, very rarely after 1000 AD) levies of free peasant farmers. There was no uniformity of equipment.

The exception would be late medieval aristocratic retinues (personal household troops), who often wore their master's livery as a sort of uniform and could be equipped with armor that their master had purchased in bulk from Italian or German armorers.

Now, that is not to say that there were not types of armor that predominated in certain places and times. The mail shirts worn by (some) early medieval warriors seem not to have differed much at all, whether that item was made in Spain or Norway. By the high middle ages, the short mail shirt had morphed into a much longer hauberk, with full sleeves and possibly mittens. Some armor types seem to have been regional. Lamellar armor - that is, armor produced from scales of iron or horn laced together - seems to have been popular in the east, but was almost unknown in western Europe.

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