How do commercial sauce companies remove seeds from tomatoes?
In a commercial setting, i.e. a food processing plant, industrial, large capacity equipment is used. This equipment is specifically designed for this purpose and is part of the production line.

In a home setting, there are a couple of options. First is an electric tomato mill. These can be a little pricey, but are designed specifically for tomatoes and other soft fruits. Some come with different size strainers to accomodate different textures and seed sizes. Here is a link to a mid-priced unit on Amazon.

Second, and more affordable, is the manual tomato strainer and sauce maker. They are much more affordable and are designed to remove the skin and seeds while leaving the juice and pulp. Here is a link to a moderately priced and highly rated unit on Amazon. The pics for this item clearly show it discarding the seeds and skins, while keeping the juice and pulp. The texture looks quite nice, also.

To be clear, I am not endorsing these items, just using them for examples.