"As we know non-adults aren't allow to carry weapons"
This is not something that we know or an obvious point. It is also not obvious that a knife would qualify as a weapon for these purposes. This would not be true in most jurisdictions in the world. I have no idea what knife control laws look like in China or Taiwan. A reference to why you think that this is the case would be helpful.
Even if there are laws banning possession of knives (i.e. carrying knives) when one is in public for use as a weapon, it would be very surprising to me if teens weren't allowed to possess knives in a kitchen, or a work site where a knife was a necessary tool. It would similarly surprise me if a teen working on a knife design in a craftsman's workshop would be illegal.
What makes you think that any of these things are illegal in China or Taiwan?
This might have been illegal in Japan in the 1600s when metal blades were highly regulated (this is one of the reasons that most Japanese food is served with portions pre-cut to be bite sized), but I very much doubt that teen possession of knives for practical purposes is illegal in any of those places today.
There is also, in general, nothing wrong with factories run by adults making knifes from other people's designs. They do that all the time and it wouldn't be illegal to do so just because the designer wasn't allowed to use the product of the factory in public. I would be surprised if a factory even asked how old the designer was, particularly if he was operating through a company formed for him (something an adult might have to do).
The harder question would be whether the teen can enter into an enforceable and valid contract with a factory without having the co-signature of a parent or guardian. Many countries don't allow this so that the teen is not exploited into agreeing to a big contract on unfair terms.
Also, many countries make a distinction between criminal acts committed by adults and the same acts committed by minors. A teen, particularly a young teen, may be under the relevant law, capable of only engaging in juvenile delinquency, rather than an adult felony. But, again, I don't know how this is handled in Taiwan or China.
In general, the legal system in Taiwan strongly resembles that of the legal systems in Continental Europe not long after World War II (i.e. in the 1940s), with its own local developments since then.
But, China's legal system is quite unique and is not very similar to the common law legal systems of England and former or current English colonies, or the civil law legal system of Continental Europe (or for that matter, Islamic law). China's legal system is different at the level of very deep concepts of legal process, of what is and isn't law, and of many core legal concepts. It also has piecemeal bits that are imposed by treaty even though they are not organically natural fits with the rest of the Chinese legal system (e.g. its intellectual property laws).