No, it's as a pp says, its a play on the medical articles in Readers Digest, always titled ' I am Jack's gall bladder' or whatever.

I read that as Jack being a popular and regular name so as he is talking about "Jack's ____" he is talking about feeling things that the rest of society feel. His experiences are not new and ge is showing lots of other people feel these things on a regular basis.

This is correct. Watch the movie, don't read the script.

That was just for easy reference for the actors. It's not his name. He got it from the Reader's digest.
Seize the moment, 'cause tomorrow you might be dead.

I repeat:
READ THE SCRIPT!

oh that makes sense, thanks!

When the narrator is in the house on paper street with tyler he finds a bunch of old books with educational articles titled "I am Jack's ____" with different body parts. For example I am Jack's heart I pump blood throughout his body blah blah blah. He thinks it is stupid and ridiculous but from then on the phrase "I am Jack's" is continually used.
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https://youtu.be/zbJjvt6bbQ8

because that's his name. He was refering to himself in a unique way.
Some people claim his name wasn't revealed but nevertheless in the script he's Jack.
just read the script:
JACK
People are always asking me if I know Tyler Durden.