This answer will apply more to the late medieval military structure in England (roughly 1300-1500). It depends on what you mean by standardization. Every soldier would be required to carry certain kinds of arms. So a mounted archer would have to arrive with his bow, sword, and horse at the very least. Ditto for the men-at-arms, foot archers, etc. The specifics of your weapon didn't matter so much, besides the obvious caveat that everything had to be in working order. There was no "regulation" lance or bow that everyone had to use. That said, we do know that some units may have had relatively standardized equipment. There are records of aristocrats purchasing large orders of weaponry or armor to issue to their retinues. It can be presumed that retinues had some degree of uniformity, but we can't say for certain how standardized they might be.

Inherited weaponry and gear was definitely seen in the ranks. We know from wills and inventories of goods following someone's death that there was a fair amount of war equipment in circulation. Now, these wills and inventories are representing a minority of people in medieval England. I'm not saying that every Englishman had a stockpile in his home. However, it is clear that among a certain group of people (the same demographic that would be recruited for military service to begin with) weaponry and armor were in circulation within families. I've seen mentions of swords, bows, lances, and bits and pieces of armor. I can't remember the article I found this in, but one man had so much military equipment stored up in his house that it's assumed he was some kind of used weapons dealer. There were plenty of options available for the soldier on a budget.