There was enough and correct said about the situation with thesis submission and formalia of the PhD awarding. I am tackling this question from a different viewpoint.
Basically, if you work at the university, you work directly under your supervisor, the university professor. Either, you are paid from state funds (which you are not), or you are paid from a third-party funding.
Most third-party fundings are basically "your supervisor get a sack of money". So, while your research topic might be shaped by the third party, your boss is still the supervisor, you work for him.
Now, I would imagine that many companies would not like this. So, in most cases of an industry-based PhD project, you work for the company. The company might be nice enough to allow you to collaborate with your supervisor on your job results academically. But this is not a given. (In this case you basically do your PhD on weekends.)
The only exception to this view is when the company plays the role of a third-party fund. So, it's basically, "here is a ton of money" for your supervisor, there is an interest for the research in the company, and there are capabilities to do it at your supervisor's lab. Where do you work depends now on some agreements between to sides, but you might have a choice.
In most cases, however, you cannot choose. You are either bounded to your supervisor (because the money for your position are with him and your university). Or you are chained to your company's chair (because they pay you and your PhD is at best a byproduct and at worst your own problem entirely).