As a general rule, company formation isn't really complicated at the small-business or founding stage. Don't add unnecessary overhead to a process where more than 70% of small businesses fail.
Analysis and Recommendations
In terms of official definition, can we consider starting a new business or a new company a project and can it be managed according ...
Welcome to PMSE!
can it be managed according to project management principles
You will certainly benefit from running it as a project. Here are some additional tips:
Clearly define the end-point: As @david-espina said, you need a clear definition of when the new business has been started. For example, you may decide that when the first sales invoice has ...
Frame challenge: You're difficult to deal with, not your colleague. (I'm assuming. Don't have enough details to see the whole picture, so this answer is a stab in the dark.)
If I were in your colleague's shoes, with you coming in to manage me (without my input, and apparently even without an official position granting that authority), and then you decide to ...
A project is defined as work that has a defined start and defined finish and has produced something of value at the finish. So any effort where you have to produce some type of finished thing in an established and defined timeframe would most certainly be defined as a project and your work would benefit quite nicely from applying PM principles to it.
I think this common mistake comes from the word "owner". "Own" has two different meanings: 1. to have, possess; and 2. to admit or confess to. The product owner is not one who have the possession of the product; rather is who admits or denies the specifications of the product in backlog.