When an otherwise vegan product has such an allergy warning, doesn’t that just indicate that it shares manufacturing equipment with other non-vegan food? Would it be incorrect to call the food vegan, even if all the actual ingredients are?
There’s no official vegan body that makes rules on such matters. There are levels of strictness within the organized vegan community as well, and individuals are free to make their own decisions and set their own boundaries.
From my experience, though, and I’ve had cause to look at multitudes of vegan products, very few manufacturers who take care to cater to vegans today would allow their products to be made on lines that would have such contamination. That was once an issue, because so many small companies would be forced to share facilities with mainstream companies. As the market grew larger in recent years, most companies have set up their own plants to ensure than no cross-contamination was even possible. That has the added advantage of being marketable to those who are allergic and want absolute safety.
Yes, that’s an answer so loaded with “maybes” that it’s not an answer. So let me come down closer to the line. Most vegans would be upset with even the potential presence of enough animal contaminants to set off allergies and so wouldn’t consider the product to be vegan. Some would just look at the ingredients and not care. I think. IMO. 
It’s perhaps only an anecdotal answer, but of the 6 vegans I know closely enough to understand what they’re particular about eating, they may avoid products with that package warning when shopping for themselves at home, but when sharing something at work, if it’s something getting passed around, as long as there are no animal products actually in the ingredient list, they’re not that picky.
This. There was a discussion some time ago about certification for Jewish dietary laws, and how some industrial production process filters something through ground bones. Now, the bones are not contained in the final product, but does that contact make it kosher or not? Most normal Jews had never heard of that production practise.
So it depends on how much the individual vegan knows about the process, or cares, or trust the company to share similar ethics vs. making a quick buck by jumping on the fashion trend.
“fashion trend?”
The truly bizarre: The notmilkman complaining about accidentally washing with liquid soap containg yogurt (oh, the horros!). He called the company to complain!
I don’t know about vegans in Western culture, but given the option Indian vegetarians will not touch any food that they know has come into contact with non-vegetarian foods or even into contact with tools that were used for preparing non-vegetarian foods.
I’m not shocked. Many vegans disagree with animal products of many/any kind being used by humans, so they would object as much to a yogurt-containing soap as a yogurt-containing smoothie. OK, this particular gentleman was a bit… unusual, but his basic question, whether or not actual milk was or was not used in the making of the soap, is something that many vegans would want to know.
Ditto with honey.
It’s stolen from the bees!
Yeah, that’s quite bizarre. It’s hard to follow his stream of thoughts, but the gist of his huff and puff is that there was yoghurt in the soap he used, which was labeled on the dispenser itself?
At any rate, an anecdotal experience for the OP. I had a few friends over last weekend, and one of them is vegan. I got some ice cream that was made with rice milk, and on the label itself, it said it was a vegan product. Then when she stopped by my place, she had brought ice cream that was made with coconut milk. It was actually the other brand I was considering, but had put it back on the shelf because it said it may contain traces of dairy. I pointed that out to her, but she was rather unconcerned, saying that they just have to put that warning there if it’s processed in the same factory where dairy was used. So for that one vegan, as long as there are no animal products in the food itself, what it touched during manufacturing does “count”.