I see three possibilities:
1) He may have had a "bad" (to him) experience with a pair of shoes other than his comfy favorites and now is locked in to only wanting his "good" shoes (common at his age). If this is the case, don't make a big issue out of it. Wait it out, don't fight. Let him wear what he wants until they no longer fit his feet, then throw away. Eventually, he will forget about the "bad" experience and forget about the "good" shoes, and you will be able to put any shoes on him you wish. Make shoe next pair of shoes is soft and has plenty of room to wiggle toes.
2) Or, the problem may have nothing to do with shoes, and is actually about control. Does this same thing happen with Dad or only with you? In general, is dad less interested in how he looks and doesn't fuss over clothes, while it is more important to you? (just asking, not criticizing in any way). How about if another child offers him a different pair of shoes? At his developmental age, control issues can come up. If it is about control, best not to fight him on it because won't help. Wait it out. Eveentually, his old shoes won't fit. Then you can get a variety of shoes for him to pick from and put on, because if this is a control issue, he will eventually want to put on his own shoes, himself. Wait it out, don't fight.
3) Or, he may be hypersensitive to tactile stimulation and really cannot tolerate any shoes other than his soft, comfortable ones. My daughter had this problem (starting around 2). Wouldn't wear socks unless they were inside out, wouldn't wear boots in the winter, wouldn't let anyone wash her hair or brush out the knots, etc. She eventually out grew it, maybe around 6 or 7 (she's 16 now). Your son can be evailuated by a health professional if you see a trend beyond the shoes.
Just my experience and opinions. You'll figure it out. Good luck!