Hi K.,
I have read all the responses so far, and I agree with those who suggested having him evaluated by a physical therapist, starting with your pediatrician. I also agree with others that say "back off a bit" and let him do it in his own time. I am now 68 years old and his difficulties sound like deja vu. I didn't get a bike until I was 8 and it took 6-8 months for me to learn, despite my poor dad running behind me holding onto the back fender for hours and hours. Historically, everything I've ever done that involved body movement and balance, took much longer to learn than everyone else; repetition, repetition, repetition; that included dance movements, skating, gave up on ice skating, standing yoga postures, tai chi, water skiing; wouldn't even try snow skiing until age 53, failed, didn't try again till age 62, and again I repeated the beginning classes several times (about some things I am pretty determined) and now have finally graduated from bunny to green slopes. The point of all this is that I think (although I can never be certain) that if physical therapy had been an option 60+ years ago, some of my frustration, embarrassment, shame, sense of failure, lack of self-confidence, possibly could have either been eliminated or at least lessened. If lack of insurance or financial means is not an obstacle to checking out the physical therapy avenue, I encourage you to look into it. If that is not something you want or can do at this time, I REALLY encourage you to allow him to let you know when he wants to practice. If he begins to see how important it is to you for him to ride the bike, that puts pressure on him to learn quickly to please you. When he indicates he wants to practice, do so very matter-of-factly, staying away from focusing on the outcome. And, of course, you don't compare him with his brother, I know, but you can bet he compares himself with bro. Try to downplay his difficulty by telling him stories, e.g. "I know about this old lady who didn't learn to ride a bike till she was 8, and she is still learning new things today, like skiing, etc."--you know, "The Little Engine That Could" type of stories.
I really empathize with your son and you. I am praying and sending him (and you) postive energy.