The reason Soda feels so strongly about Mickey Mouse comes from a sense of connection. Soda feels out of place in his world. He is expected to attend school, but he hates it; his brothers, on the other hand, love it and are good at it. Soda eventually drops out, because he knows he won't succeed there. Soda is the middle child, stuck between the warring Darry and Ponyboy. Darry is the protector, Ponyboy is the protected and Soda is ... out of place. Mickey Mouse is an ornery horse, not well-liked by other horses and most humans. He is out of place in his world, and Soda sympathizes with him.
Mickey Mouse belonged to a guy at the stables where Soda used to work. He was an ornery creature, but Soda loved him and Mickey Mouse loved Soda back. Soda went to see him every day, and the normally difficult horse responded by being gentle with him and coming when he was called. As Ponyboy put it, "he may have belonged to another guy, but he was Soda's horse".
After awhile, the horse was sold, and Soda was broken-hearted, and "bawled all night long". Ponyboy summed it up well - "Soda never really wanted anything except a horse, and he'd lost his" (Chapter 3).