I thought that people who manage their business alone is called "solo business owner". A Japanese news site says that "own my own business" is the proper word. I feel it strange. May I ask how do native English speakers say the type of business owner in English?

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‘Solo business owner’ is unnatural and unclear.

‘Owning your own business’ doesn’t mean that you work by yourself and manage the business alone, though it could be the case.

A ‘solo practitioner’ is often a professional who is the only professional in the business, but that also doesn’t mean that they work alone or do the management by themselves.

‘Self-employed’ and ‘sole-proprietor’ are terms related to US tax law but don’t define the details of the business.

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a person in the newspaper should be fired for using the tautology

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'Own my own business' is like a sentence fragment. 'I own my own business,' or 'I have my own business.' For second person: 'to have one's own business.'

'Solo business owner,' 'sole proprietor,' 'owner and operator,' etc. are noun phrases.

Answer

Hello Yoichi,

I own my own business. (most common but doesn't automatically mean working alone. You can be an owner but still have employees)
I'm the sole proprietor. (Sole Proprietor is the most formal legal term)
I own a sole proprietorship.
Solo business owner is more informal in my opinion.
Hope this helps.