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Following situation occurs/ed. I meet a friend for a coffee. We chat and I ask him what he's doing for work now.

His answer: "I'm teaching English."

This irks me. Because in that situation he's not teaching right that moment, but rather he teaches English. So shouldn't his answer rather be "I teach English."?

English is not my mother tongue but I do think I'm fluent in it. I grew up bilingual.

Q: What are you doing at the moment?

A: I'm teaching English at a language school.

Does this mean the person is teaching at the moment of speaking. No, it doesn't. We use the present continuous tense to talk about things that are in progress or for actions that are, for the time being, temporary in nature. The fact your friend replied using the present continuous means he does not consider teaching to be his permanent job, it is a stopover, something he is doing now, for a limited period. Of course, he could always change his mind and become a full-time fully fledged teacher, but his answer was grammatical and perfectly acceptable.

Wikipedia

The present continuous is used in several instances.

• To describe something which is happening at the exact moment of
  speech:
  The boy is crying.




• To describe an action that is taking place now, but
  not at the exact moment of speech:
  He is working in Dubai.




• To describe an event which is planned in the
  future:
  I'm resitting my French exam on Tuesday.



• With always, but meaning often:
  My mother is always making me go to school!
  She is always playing with that doll!
  

It depends on the circumstance.

If you are actually in a classroom, teaching an English class and a student asks:

"What are you doing right now?"

A correct response would be:

"I am teaching English."

The response describes your current activity.

If, however, the student asks:

"What is your occupation?"

The correct response would be:

"I teach English."

This response describes your occupation.

Two different questions, two different answers.

It is the difference between present progressive and simple present.

"So what are you doing nowadays, Don?"

"I am teaching English."

You taught English yesterday, now, and tomorrow.

As to your annoyance:

Your understanding of the progressive tense is that you have to be doing the thing right that very second, but that would mean we would never say "am running" unless we are on the phone panting because we are running while talking.

As you can see, the posed question asked what his profession is and was, what he has been doing, so, at this very moment, and in the past, and in the future, he was, is, and will be a teacher of English.

"What do you do, Don?"

"I teach English."

Simple present--right now, I teach English.

As for "to teach", "to teach" is an infinitive that can act like a noun, adjective, or adverb.

"I plan to teach English." (Direct Object)

"To teach English is my goal." (Subject)

Your friend answered "I am teachING English"simply because you asked "What are you doING." If you had asked "What DO you do now?" Your friend's answer would be "I TEACH English."
Please remember your friend IS teaching English (at that time). That's how he teaches his students to answer questions.