Did Sigourney say "Weyland-Yutani"? I don't think she did. Don't make assumptions, you silly person.
"The Company" could just refer to the company in that particular film. It doesn't matter if it's a different company, or that it's military rather than corporate, it's still a company. Stop being a nerd, get laid, and stop caring so much about the details of a sci-fi franchise.