I'd like to back this up as I work in food manufacturing (in the UK). The majority of food companies do not have the funds or turnover to have their own dedicated factories - I would wager most decent-sized vegan companies share production space with non-vegan (whether they have dedicated areas/equipment or not) but law mandates that, at least for allergen purposes, equipment must be cleaned thoroughly between product changes. Cleaning in food manufacturing is pretty hardcore, too.

It's also worth noting that, in the EU anyway, gluten is the only allergen with an upper limit on the amount allowed in a "gluten free" product (hint: it's not 0, but it is 20ppm) which needs full lab testing and certification. So dairy, for example, requires testing but has no upper limit and is usually verified by swabbing equipment which may not be as sensitive (but still in the ppm). Essentially, you will never ever be 100% certain that any food is totally dairy-free, BUT the numbers will be absolutely minuscule (unless there's been a fuck up, which is when you get recalls).

This doesn't cover restaurants or eating out, of course, but can give you a picture of manufacturing. I'm happy to eat pre-packaged vegan food, anyway, and the more vegan products people buy the bigger the industry will get and segregation of products will grow!