I don't have an answer to knights of the medieval period in particular but the training and relationship between horse and trooper in the Napoleonic Wars and beyond is essentially that first and foremost that it was a primary mode of transport and fighting platform. In a sort of quid pro quo scenario, a trooper that looked after his horse and that his horse was attuned to could make your life on the trail/campaign much easier. A horse soldier without his horse is not well equipped and trained for life on the march, although towards the end of the Napoleonic Wars, the Grand Armee had some dismounted dragoon units (distinct from what the British called dragoons, which was every mounted unit in existence). A late as the Australian campaign in the Middle East in WW1, the reports of troopers loving their mounts and the great care and dedication which the men made for them are as common as accounts of general staff aides riding horses to death to bring news and/or messages between generals.

Certainly, a lot of famous generals and marshals had horses which they loved; the Duke of Wellington had a horse which he rode throughout his Peninsular Campaign and Waterloo (a half Arab named Copenhagen) and the Emperor Napoleon had an Arab which he rode from his Egyptian campaigns all the way to Waterloo. Both of these men had a great love of their horse, I guess very much in a way you have a favourite dog but I wouldn't like the speculate what sort of affection they had to their mounts - they certainly loved them, after a fashion. The hunting dogs of Louis XIV of France were greatly loved as well - there are paints and bronzes of these guys all over Fountainbleu.

All in all, the conclusion I would draw is that a knight would love and care for his horse so long as it was able to fulfill his or her obligation as either a war horse or palfrey, for which either were expensive commodities which affected a knight's ability to fight on campaign much more than, say, the quality of his arms and armour but less than his retinue of sergeants and 'staff' for want of a better term.

Sources:

Australians at War (?) - Unknown (pre 1950), Monash University Library Collections This is not a properly cited source, I'm sorry! It was missing pages and water damaged to boot!

Captain Coignet - Jean-Roche Coignet (Published Leonare 2007)

Swords around a Throne - J R Elting (1988)

Breaking and Riding: With Military Commentary - James Fillis (2005)

Interesting read: https://www.awm.gov.au/wartime/44/page54_bou/