wrote:
In a recent flight, I received the handoff, "27D change to Chicago
Center on xxx.xx" Next call, "Chicago Center, Arrow 27D, 3000"
Response: "Arrow 27D, this is not Chicago Center, this is Chicago
Approach". Me "Arrow 27D, correction noted"

Is this an issue of sufficient importance to waste the air time in a
very busy airspace? I'd just like to know...

It's his air time. He can waste it at his discretion.

If you were handed off, he knew you were coming, he knew you were on the right
freq. Not clear what his motivation was in making the correction, but it really
doesn't matter why he chose to make the correction. Just acknowledge and press
on (as you did).