They didn't figure out anyone's ethnicity, according to their press statement they banned all accounts which were accessed via Iranian IP's. While seemingly a little heavy handed IMO, they are trying to comply with US laws. They also provided a way for people to send an email to address their specific accounts if they feel they were terminated improperly.

Slack is protecting their multi-billion dollar business and their investors which are both based in the U.S. If they do not, they could run into trouble and put far more at risk. This does not only affect Iran, it affects other countries too which the U.S. has active sanctions against.

Not just a little heavy-handed. If that's what they did, they probably picked up users that just happened to use the service from one of the countries at some previous point in time. And, at that point, the sanctions may not even have been in place.

Now, I admit that visiting the countries on the list aren't on my to-do-list, but if for example I decided to make a vacation trip to Cuba I really shouldn't lose my account at any US-based service just because of that.

Which is a good argument for cutting down reliance on American companies.

That's a question for Slack how they found out and that is a great question, but there seems to be much conflation of that question with justification of sanctions in the first place, which I wanted to address. It's a question of nationality and citizenship if Slack determined one of those to be the case then they simply followed the regulations set forth by the sanctions.