First of all, there is no reason that your child needs to "give up Chinese citizenship" at any point. The idea that countries "allow" or "don't allow" dual nationality is a misconception. Countries just have various rules for gain and loss of their nationality. Each nationality is separate. According to Chinese (PRC) Nationality Law, if your child was born outside China, and your wife did not have foreign permanent residency (in Japan or another country) at the time of your son's birth, he is automatically a Chinese citizen at birth, even though he has another nationality at birth. If he was born in China, he also he is automatically a Chinese citizen at birth, even though he has another nationality at birth. Chinese law does not provide for that Chinese citizenship to be automatically lost unless he voluntarily gains a new nationality later in life.
However, you probably got the wrong document for him, as the current usual practice for Chinese dual-national children is that they don't get Chinese passports. Instead, the usual practice is:
- When they are abroad and need to travel to China, they would go to a Chinese consulate and apply for a Chinese Travel Document (a passport-like booklet valid for 2 years and identifies them as a Chinese citizen), which they can use in conjunction with their other passport to enter and exit China for the duration of its validity.
- If they are in China and the Travel Document is expired or lost and they need to travel abroad, or they were born in China and this is the first trip out of the country, they would go to the Public Security Bureau and get an Entry-Exit Permit, which they would use with their other passport to exit China. When they need to return to China again they would get a Travel Document as above.
I am not sure why the consulate or agency even issued him a Chinese passport in the first place. Perhaps they were unaware of his multiple nationality. If he was born abroad, the consulate may have missed it because the dual nationality was with a country different from the host country. For example, Chinese consulates in the US regularly issue Chinese Travel Documents, because children born in the US automatically have US nationality, so Chinese passports are out of the question. However, assuming this was the Chinese consulate in Japan, they may have only considered dual nationality with Japan (which requires a Japanese parent), and since this is not the case, they failed to consider dual nationality with a third country.
Having a Chinese passport may cause him problems when leaving China, which I think you may have realized from your questions. There have been many reports that children holding a Chinese passport (e.g. they accidentally got a passport in China after being added to hukou) and a foreign passport are denied exit by the Chinese border control. However, Chinese border control have no issues with someone holding a Chinese Travel Document (or Entry-Exit Permit) and a foreign passport. (Not sure why there's a difference, as both the Chinese Travel Document and Chinese passport identify the bearer as a Chinese citizen.)
The reason they find out is because Chinese border controls, for some weird reason, checks whether someone leaving the country with a Chinese passport has a visa or other document that allows them into the destination country (or that it's a country that allows Chinese passport holders visa-free access), or else they will not let you through. (Not sure why they care, as it is not the Chinese government's problem if you can't enter another country, only the airline's; so if the airline lets you through, why does the government care?) And most of the time a dual national will enter the destination country using their foreign passport.
From your questions, you seem to be aware of this potential problem and are seeking the solution of entering the destination country using the Chinese passport with a visa for that country (since luckily it isn't a country of his nationality), and hiding his foreign passport. That could work, but it's a lot of hassle and you would have to do things consistently (e.g. enter and exit the UK using the Chinese passport, or in the future they could wonder why there are no stamps from the UK, etc.), and in general, it's not preferable to have to hide things from border officials.