It's difficult for me to speak broadly about Europe on this topic, so I am going to talk specifically about English soldiers in the Hundred Years War between 1415-1453.

In the most general terms, there was obviously the tournament as a source of knightly training as well as a standard military education for boys of the upper social classes. These men would certainly have been well trained - a good example of this standard of training is quite famously the mock-retreat at the battle of Hastings.

One of the more famous examples of widespread training were, of course, the English longbowmen of the Hundred Years War. Essentially, they were legally required to practice regularly, creating a large pool of skilled man power that could be drawn upon for the wars in France. See the 1252 assize of arms and Edward III's later declaration on the subject: "Whereas the people of our realm, rich and poor alike, were accustomed formerly in their games to practise archery – whence by God's help, it is well known that high honour and profit came to our realm, and no small advantage to ourselves in our warlike enterprises... that every man in the same country, if he be able-bodied, shall, upon holidays, make use, in his games, of bows and arrows... and so learn and practise archery."

In general though it's a mistake to think of medieval soldiers in terms of an official national military. Something like a professional class of soldier begins to emerge in England during the later stages of the Hundred Years War, as the need to continuously garrison northern France creates a situation wherein soldiers drawn from all levels of society begin to serve serially in France, often under the same commander and specialising in a particular type of service.

So, for an example of this phenomenon we have the archer John Powcher. Powcher served in the Harfleur garrison in 1423, 26, 29, 31 and 35 and nowhere else. Powcher also served consistently under a single captain - William Minors, the commander of the Harfleur garrison. Another archetypal example of the English soldier of the latter stages of the Hundred Years War was the man-at-arms Thomas Scolles. Scolles served under Lord Willoughby in France extensively and specialised in field service - only serving in a garrison in 1434 at Pont de l'Arche. These early 'professional' soldiers are fairly common during this period; whilst they're nowhere near a majority a number of them appear in a great many garrisons and field retinues during the period.

To conclude, we have fairly good evidence for a high level of military expertise in some parts of Europe towards the end of the Medieval Period. But the kind of massive training exercises you're asking about? I have never encountered evidence for them.

Sources:

Curry, 'Guns and Goddams: was there a Military Revolution in Lancastrian Normandy 1415–50?'

Bell, A.R., Curry, A., King, A., Simpkin, D., The Soldier in Later Medieval England

Details of soldiers careers taken from medievalsoldier.org.

For general stuff on knights, chivalry, the tournament, all that jazz see Nigel Saul 'For Honour and Fame: Chivalry in England, 1066-1500'