Nephew Is Very Shy and Emotional

Perhaps, the whole family needs help... Counseling or parenting workshops.

One thing is:
1) this can be due to personality, or
2) it is because of how he is treated and what 'expectations' the parents have upon him. Thus, a child is then 'afraid' to do things, because of it.
And it creates anxiety/fears in a child.

To me, his Dad being so rigid, does not help.
At all.
His Dad, denies his son has a problem.
His Dad, is 'very strict' as you said.
Thus, the son probably did not 'learn' that he can communicate and express his feelings etc. And that in itself, can create an insecure child.
There needs to be bonding, and that the child knows his parent accepts him for who he is. That it is unconditional.
Boys... NEED to learn, that they can express themselves. Good or bad, and that the parent is there for them. Not that parent just scolding or judging the child.
His Dad, having that 'view' of his son.... will just create a child that is pent up and insecure and does not know himself. THAT, is the how kids become... sad/insecure/fearful.

The boy, needs to know himself, feel good about himself, be respected for himself and what interests HE has and that he is allowed to be, himself. Not just being at the end of a rigid set of expectations.

To me, the boy is probably more comfortable with his Mom.
Not his Dad.
His Dad... unnerves him and deprives him of confidence.

Have they considered having him evaluated? What does their family doctor/pediatrician think about his behavior/personality? His reservedness by itself might not mean anything at all. But if he is crying daily over having to be in a setting outside his home (daycare in both locations, karate, etc) and he sits by his parents at birthday parties when he is familiar with the setting and all the people there, then it sounds like there is more to it. The not eating unless mom cooks it is odd, too. Does he eat store bought cookies?

Do his parents seem at all concerned about his behavior?

I was thinking the same thing as momto2princesses, read that bood and search for others like it. My daughter is very sensitive, but acts out in the opposite way, my friend has a daughter who sounds very similar to your nephew though and she has read all the books on highly sensitive children, and has adjusted their lifestyle to help her daughter with this. Maybe

His parents should probably consider a referral to a therapist or someone who specializes in children's mental health issues - he might have some form of anxiety or other emotional problems, and there is help available, but the parents need to be willing to seek it out. Do his parents perceive his behavior as being a problem or do they think it's something he needs to outgrow and are they okay with it for now?

IMHO professional advice is needed here . . . and possibly some family counseling (to discern if there are any family dynamics exacerbating the problem).

With the OCD-ish behavior (extreme particularity) I would also research PANDAS (make sure he doesn't have strep lingering in his system).

I would also seek out an integrative physician to see if there could be a physical component to his anxiety. A neuro-psychiatrist might be a good resource too.

Wishing lots of luck and good health - sometimes we moms have to be detectives!

Start by having his mom/dad read this book:

The Highly Sensitive Child by Elaine Aron

Also might be worth asking the pediatrician what he/she thinks.