How to map various buildings belonging to the same company?

I agree that name=company is a bit strange, but I'd go farther and say that it is, in fact, incorrect. It's tempting to tag things so that they are rendered how you or I'd like. This is called Tagging for the renderer, and is well established as an error. Your alternative suggesting the operator tag is the correct approach, even if it's not as gratifying when you don't get to see it in the default rendering. Do remember that the default rendering is just one of many. If someone wanted to have a map showing all the operators of buildings, and you'd put the operator in the name tag, then your buildings operator wouldn't show up on their map.

As has already been suggested the operator's name could be put in an area tagged as landuse=commercial (or landuse=industrial, or potentially another landuse) tag. However this should only be done if it's a reasonable name for the area. For example, if there are many buildings belonging in an industrial park owned by Bob's Bicycles, and the industrial park called Bob's Bicycle Industrial park.