How can I tell if some enterprize is an authorized apple service provider?

You are correct in being suspicious. Do not proceed any further with this company. They are phishing for information and/or trying to convince you to take actions that will give them control of your system.

This article from Apple provides additional information.

Avoid phishing emails, fake 'virus' alerts, phony support calls, and other scams - Apple Support

The above article suggests contacting Apple in your situation. Here's how you can contact the real Apple support team.

Contact - Official Apple Support

Additionally, FYI, there is absolutely no one in the world (including Apple) who could possibly know what is on your computer unless you had allowed them physical access to it - so, any pop ups, emails, etc are bogus.