Is it legal for a Private Pilot to fly his own aircraft (owned by the pilot) to perform aerial surveying/mapping for a company that he owns? He sells the data to another company that he also owns.
This is probably illegal. In the Perry interpretation, the FAA considers a similar situation and decides it is illegal.
In order for aviation to be "incidental" to business, you need to remove aviation from the business and see if that changes the fundamental character of the business. If the data you're selling is something you could collect just as well without an airplane, you might have an argument. Certainly selling any of the actual photos taken is verboten.
The answer is, it depends.
14 CFR 61.113, Private pilot privileges and limitations: Pilot in command, states:
(b) A private pilot may, for compensation or hire, act as pilot in command of an aircraft in connection with any business or employment if:
(1) The flight is only incidental to that business or employment; and
(2) The aircraft does not carry passengers or property for compensation or hire..
I tend to favor the most permissive interpretations, and would be inclined to say that you could do this on a Private Certificate, but the legalities would hinge on how your characterize your business.
For example, if your business model is interpreting, processing, packaging and selling a data product, and the raw data could also be gathered by means other than flying, then one could argue that flying is incidental to collecting the raw data. (saving hours or days of driving - just like flying vs driving to meet a client to sell life insurance is incidental to the act of selling.)
If, however, you market yourself to the public as an on-demand aerial survey company you would put yourself in a very different legal position. If you have concerns about this you would be wise to consult with an aviation attorney.