The employer requested one of us to make a contract in our own name for personal unlimited wifi and use this for working purpose instead.
There are many reasons why your manager could be suggesting this. Not all of them are inherently malicious.
One well-intentioned possible reason would be that the company's internal process is too slow to actually solve the issue in the short term (approval process, product acquisition, ...), whereas reimbursements are at the manager's discretion.
Another reason is if the manager assumes that there will be personal use. As an example, my employer doesn't give us company phones because we hardly ever need them professionally, but they do reimburse us when we buy a phone (up to a given budget - we can go over the budget at our own cost). The phone is our property, but the company pays for it (partially) in order to justify requiring us to have a phone (they do pay for the provider contract, however).
To me, this makes sense from a financial perspective. It'd be more expensive to purchase phones (and support them), and most usage of the phones would be "wasted" on personal usage anyway.
I can see how a manager could assume that personal pocket wifi will similarly be used personally more than professionally. That doesn't mean they're right; but it's not malicious to assume so.
The request for you to do so, even if not malicious, is still unusual. The best approach here is to ask for clarification from your manager. Ask why you'd have to sign your name to the contract instead of the company doing so.
If there is a justifiable reason, e.g. the company can't provide you pocket wifi before the current project deadline, I (personally) would be okay with temporarily using the workaround.
If there are legal considerations here (you mention "personal" unlimited wifi - is there an issue if you use this professionally?), make sure that the agreement between you and the company is documented.
If there is no justifiable reason, or you simply don't feel comfortable doing so, you are legally able to refuse doing so. This can possibly harm the relationship between you and your employer, but that depends on how they handle your refusal (and also how you phrase your refusal - "Fuck no!" would be an obvious example here).
Legally, you can refuse to do so.
I hope this is not stereotyping too much, but as far as I'm aware, Japan has a cultural habit of having additional expectations of their employees. The example we hear about the most is being required to stay at work longer without being paid overtime (again, correct me if that's a wrong stereotype).
If there is a cultural expectation to cater to the company's request, you might be less able to refuse; but I simply can't answer that as I have little experience with Japan's work culture.