This is a lovingly broad question, and the sources for how knights felt about their horses varied not only between periods and geographic regions, but also within classes and even from person to person. A knight in Chinon might see his horses as nothing but a tool, while his neighbour might view them as his companions in arms.

However, I will refer to the Anglo-Norman knight and use as a case study to display what I think might well have been a common attitude towards horses in the English, Norman, French, and Breton military culture of the time.

I will use William Marshal as a good example of a popular figure we think very well might have seen his horses as not only key pieces of equipment, but companions and friends.

But first, I'll clear up your last question; the different types of horses. A palfrey is generally a light, durable riding horse. Not too big, not too expensive, but a good mount you could comfortably ride at a gallop if unarmoured. These are the ones you would ride for long distances, and tended to not be very big or warlike. A destrier is a horse-type that has attracted a great deal of debate; I can only put forward my own views on the subject. Likely, a destrier was a very stocky, very powerful horse between 12 and 15 hands high, trained to kick and bite other horses and foes. It could carry an armoured knight, weighing perhaps as much as 250 lbs, at a gallop. I come down on the view that medieval warhorses were trained to make contact, so this informs my view.

Anyway, William Marshal was known for often acquiring the horses of other knights and lords in tournament. From the Histoire de Guillaume de Marechal, we have the quote:

'He swiftly stretched out his hand towards a horse from Lombardy, and its rider was not sufficiently bold as to dare to defend it.'

It is a useful note to say that Lombard horses were particularly valued. Another quote showing William's predilection for acquiring horses is shown in the same source, after seeing a noble dismounted, William apparently;

'quickly took his horse's bridle and rode off towards the men on his side.

Clearly, Marshal valued the mounts highly, willing to strike at an opportunity for such a mount when presented. However, did he see them as anything more than a valuable possession? We have no real way of knowing. The iconography pointed out in 'The Medieval Warhorse' by Ann Hyland indicates that the warhorse was a fundamental part of a warrior's identity in the High Middle Ages. The nobility were dominant upon the battlefield because they often trained for huge stretches of time mastering mounted combat. Knights would spend hundreds of hours training and working with their preferred mounts. It is reasonable to state that a knight without his horse was seen as less noble, less prestigious, less dangerous, and likely less holy than one in the saddle.

William Marshal certainly valued his destriers highly. Upon coming across his wounded destrier, he seemed to have been pleased by the superficial nature of its injuries;

'The Marshal came to look and together they found seven wounds on the horse's body, made by the steel-tipped lances. There were wounds to the shoulders, neck and chest, but the Marshal was in no way concerned by this for he saw the horse would make a good recovery.'

Can we thereby infer a relationship similar to that shared by many riders and their mounts today? Maybe. While the evidence is hard to find, we must also look to modern tribesmen and those who depend upon horses for their livelihood; almost all of whom find great affection for certain individual horses and less so for others. We know William Marshal was a very talented horseman, and we know he likely rode almost daily for the entirety of his adult life.

There are occasional sources for affection towards mounts by lords, monarchs, and knights, but from these I hesitate to make sweeping generalisation about any entire social class. However, I would strongly suggest that a large proportion of medieval knights valued their prized warhorses (or coursers, or chargers, or destriers) extremely highly, while viewing less expensive mounts as not quite so vital. In turn, I would suggest based on the iconography featuring horses, the writing of authors like Geoffrey Chaucer and Anna Komnene that punctuate the high value knights placed upon their horses, and the observations to be found in the modern world of similar relationships, that many knights had great affection for individual mounts, seeing them not only as pets, but as friends and trustworthy companions.