OT imo hard to beat The Complete Angler

Fishing/Angling is our life

While from M-W, it appears that angling is indeed restricted to fishing for pleasure, the word "fishing" can also be used/is used to mean doing the same thing for pleasure. In my experience, in everyday use of English, "fishing" is more common than "angling."

Yes to hbberlin.

n.b my suggestion - variation on The Compleat Angler

Angling for (a) life... seems to be what many are doing...

While "angling" is definitely known in the US, I wonder if it's used more frequently in BE. OT re #1: It's actually "The Compleat Angler" if I recall correctly. Shows its age. /OT an·gling noun \ˈaŋ-gliŋ\ : the activity or sport of fishing for pleasure

fish·ing
noun \ˈfi-shiŋ\

: the sport or business of catching fish


fishing
noun (Concise Encyclopedia)

Sport of catching fish—freshwater or saltwater—typically with rod, line, and hook. Fishing is as old as the human ability to use tools to capture prey. The first significant modern innovations, including use of a reel, a rod with line guides, and a hook with an offset point, came in the late 17th and early 18th centuries. Horsehair was used as line until the mid 19th century, when it was replaced by textile materials; these in turn were replaced by nylon in the 1930s. Wood and bamboo rods yielded to rods of fibreglass and other synthetic materials. Forms of sport fishing practiced today include fly fishing (freshwater), in which a fly-like hook is repeatedly cast upon the water surface to attract biting fish; bait fishing (fresh- and saltwater), in which live or artificial bait is set or drawn below the surface; and big-game fishing (saltwater), in which heavy-duty tackle is used to land large marine species (including tuna, marlin, and swordfish) from a motorized boat.

#5 I opted for The complete Angler, which I knew would be corrected swiftly ... since we are somewhat past Cromwell and the Restoration...

An angle is a hook, so while all angling is fishing, not all fishing is angling. Also I would agree that usage dictates that 'angling' is a leisure activity.

...since we are somewhat past Cromwell and the Restoration...
which is about the era that I put the term "angling" in. ;-)

I'd suggest "We live for fishing/angling".

Anglers are usually those who are rather dedicated to what they do, in my understanding, and for fun. The ones who take it more seriously and get annoyed when boats go past. Fishermen catch for a living.

Perhaps in BE that distinction is taken more seriously. (The entries from the two AE dictionaries I cited clearly indicate that "fishing" can refer to the sport as well as the business--at least in AE.)
My dad went fishing every summer for a week or two. I sometimes went with him. We fished, we didn't angle. Given the few fish that were caught, if my dad had fished for a living, we would have starved.

#10: we didn't angle

That part of speech sounds distinctly odd to me (unlike angling or angler).

It sounded odd to me, too, but Collins accepts it:
angle2
Definitions
verb (intransitive)

to fish with a hook and line

I agree with KinkyAfro: I would use angling/angler but never angle as a verb in that literal meaning (I would use angle in a figurative sense: What are you angling for?).

Warum sich nicht ganz vom Dt. lösen, klingt mir eh ziemlich pathetisch. Vorschlag: hook, line & sinker

Ditto #8. Izaak Walton is dead and buried and doesn't influence modern English, except in the archaic spelling of the 'Compleat Angler.' I would consider 'angle' in this sense equally obsolete or archaic (or British?), surviving only (barely) in the figurative sense 'angling for compliments' (but 'fishing for compliments' is far, far more common).

For a modern company, I would say 'We live to fish / We live for fishing / Fishing is our life / etc.'