Is it illegal to have double coverage medical insurance. For example; can one person have two Kaiser plans so they don't have to pay any copays?

You may check with the company, but me and my husband have two policies through Anthem one through his work and one through my job. It just happens that both places offer Anthem. We have had both for over Four years and never a problem, and yes it does help we usually have no co-pays and very little medical bills. I would check with your insurance company to be sure. Hope this helps and please remember this is just my opinion.

You can not have two of the same plans, but they sell plans that cover your deductables. So you can buy separate plans that will work together
But starting last year, there were new policies developed, that is in about 2/3 of all of the US states now, that do nothing but pay your deductables and co/pays

My policy specifically outlaws double coverage. My wife and I cannot have two plans if we work at the same employer.
But if she works at a different employer which would be a different contract (could be same insurance company) the contract says:
If the claim is for me, the claim goes to my insurance carrier first. If the claim is for her, the claim goes to her carrier first.
After my carrier decides what to pay for my claim, any balance (that they don't pay) can be submitted to her carrier for possible additional coverage, but between the two companies they won't pay more than allowed by either company. Her claims go to her carrier first etc.
Say a procedure (out of network) for me costs $ 1,000. Submit to my company, their max. is $ 800 so I submit the $ 200 claim (balance they didn't pay) to my wife's carrier. Their max for that procedure is $ 900. They will pay $ 100 of my balance and this case I will pay $ 100 because the Dr. I used does not participate in my plan.
One other note is in the same section of the contract: The National Association of Insurance Commissioners (NAIC) has this rule for Coordination of Benefits: "The parents' birth dates determine the coverage of the children covered under both parents' plans when the parents are not divorced or separated. The responsibility for primary coverage falls to the parent having the earlier birthday in the calendar year."
I can't say for certain about your Kaiser plan but if they work like my plan, mine can't have spouses with separate policies under the same plan (basically same company/same plan). There again, with mine if it is different employers for spouses, they can have their own group plans. But with my plan rules applied to your situation, if both had a $ 50 copay I wouldn't get any additional co-pay coverage, but if one was $ 100 and the other
$ 50 you might get the $ 50 back (might cause I don't know).

I have always been on my husband's insurance. This year I signed up for insurance through my own company just because some of the benefits are better than my husband's which is HMO. The problem is now my own insurance has become primary with $30 office co-pay and my husband's is my secondary with only $15 office co-pay. I just found out that I have no choice but to use my primary insurance first and pay the higher co-pay. Is this true? Does anybody on this forum know?

Thanks

Yes, it's true - I'm amazed that you are allowed to carry two health insurance policies for the same treatment(s).

In NY (where I am) you cannot carry two policies. In fact, you are asked if you have any other health insurance when you apply.

Can I have two Kaiser insurance coverage for my child

Probably. But it is doubtful that it would be to your advantage. If you submit a claim to one, and there are co-pays or deductables, the other policy is not obligated to pay them either.