No, I don’t think it’s reasonable to classify knights as militiamen. But given the facts you’ve stated, I can see how you might reach that conclusion. My impression is that you associate being a professional soldier with serving in a standing army, and anyone who does not belong to such a force is a militiaman. I think that’s an overly simplistic bifurcation. Rather, I would say that a man (or woman!) is a professional soldier if fighting, or preparing to fight, is his primary occupation.

The stereotypical citizen soldier of classical Greece or early republican Rome was not, primarily, a fighting man. He possessed an occupation - farmer, tradesman, aristocrat - outside of war. When called upon to serve his community, he took up his arms; when the campaign was over, he went home and resumed that occupation. While not under arms, he had other responsibilities that occupied his time and attention. /u/iphikrates can tell you better than I, but my understanding is that most Greek hoplites engaged in very little in the way of peacetime training; certainly, they undertook very little, if any, organized drill.

The knight, in contrast, was a member of a self-conscious military aristocracy. He was socialized into the profession of arms from an early age, growing up around and among knights and lords, and began training for his future career when he was old enough to sit a horse and grasp a practice sword. His self-conception and identity were bound up with fighting. He had no other calling or marketable skills, unless you consider landownership a profession (I don’t, and even then, there were plenty of landless knights until quite late in the period.)

Knighthood, as an institution and a class, came into being following the breakup of the Carolingian Empire, during a time of great political disorganization and turmoil. This was a period of fairly frequent war between kings and great lords, but it was also marked by endemic private warfare between petty lords and castellans. Knights might be called on to fight because of vassalic or familial ties, but they might also voluntarily participate in hopes of winning land, reputation, or a wealthy wife. For these military professionals, war was not only a matter of duty, but an opportunity for social and economic advancement.

While it is true that medieval armies were largely ad hoc affairs, assembled for a campaign and then disbanded upon its conclusion, the knight did not cease to be a fighting man just because there was no war available! From as early as the 11th century, tournaments - at this stage, brutally violent “mock” battles rather than the formalized jousts of the 14th-15th centuries - occupied a considerable portion of the peacetime energies of the military aristocracy. By the 12th century the tournament circuit was fully established, and knights, especially younger sons with no land of their own, might travel hundreds of miles and fight in a dozen or more tournaments each year. William Marshall, the future regent of England, built his reputation in just such a way, eventually becoming a household knight of Henry II and then Earl of Pembroke.

I believe you're missing two key elements: intention and training.

Firstly, intention. With the exception of the Spartans, the leisure classes of the Greek city-states were not intended to be warriors. While there was the expectation that they would serve as heavy infantry in a war, that was only incidental, not the whole ideological reason behind their wealth and free time. /u/Iphikrates will be able to give a more certain answer, but my understanding is that owning arms was not technically a requirement of citizenship, merely providing them for yourself when needed. Further, not all of the citizenry were required to serve as hoplites and most of those who did fight as heavy infantry were not exactly a part of the leisure class. Light infantry, marines and oarsmen were generally drawn from the less wealthy citizens of the City. I believe it was much the same for Rome.

In contrast, the whole purpose of a knight was to fight. The Latin term for it, miles means soldier, and it became increasingly connected with the knight as the medieval period progressed. Knights were the second of the Three Estates, the bellatores. While the priests prayed for mens' souls and guided spiritually, and the commoners worked the land and labored with crafts, the knights defended the other two Estates (theoretically) and ruled them temporally. Their entire reason for existing was to fight.

The second element is training. As /u/Iphikrates has written before, there is no evidence that any of the Greeks, including the Spartans, trained for war. Further, it is possible that there was an ideological resistance to training for war.

In comparison, knights were expected to train for war. The degree to which they actually practiced this ideal varied on the time, location and length of relative peace, but it was what they were meant to do nonetheless. It took considerable training to use a lance on horseback and to fight in formation - whether in the small conrois of the early period or the massed charges of the later era. A big part of this training was the tournament, especially in the 12th and part of the 13th century, where tournaments resembled less lethal battles and both knights and their footsoldiers participated in them. Even when the later tournaments focused far more on individual combat, this still in large part focused on horsemanship and control over the lance.