So, I kind of work in this space. On websites you can have prompts or call to actions that get triggered after a certain amount of time or scroll position or page being looked at etc. These will show a "new message" in the chat bubble to say "can we help you with anything?" I don't particularly find this invasive but I do find it annoying as I do usually know what I'm looking for when going to a website. However, I know, as you agree, companies use these techniques to upsell and gain increased online sales. And they work.
Then you raise a second point about emails. This would be completely different as the business in question would be contacting/spamming you directly. This scenario will soon be covered by GDPR, in the EU, so this will be less of a concern as of 25th May 2018 for EU citizens. I don't know think the prompts in the first case would be affected by GDPR, as long as the data isn't stored anywhere, but it might still fall under the "processor" business type covered in GDPR.
I guess there is also a psychological aspect to this as well. If you prompt users during their "session" then it can provide instant gratification as the visitor doesn't have to do anything, and the chat bot/agent will do it all for them.
tl;dr I think their is a balance, if the prompts are intelligent enough to know when, 9 times out of 10, a user is stuck then all good. But if a company is using it these kind of tools without any thought to the end user then they need to re-evaluate how it values its users.