Braiinz is the company that runs slushpool.....Braiinz OS is just a product they have put out. Is it really that confusing? Many many companies have a corporate website that is different from the website they offer products through.
I don't doubt that they are the same company, but I think grinbucks point is how can we be sure its actually the same company.
We can be sure it is the same company through a number of methods:
1.) There is a press release announcing the release of Braiins OS on the SlushPool website.
2.) The OS release is signed by a GPG key found on the SlushPool website.
Given these factors I would say that yes, it is the same company behind both SlushPool and Braiins-OS.