I have four of those 2' x 3' patches and in all cases the new drywall is
always about 1/8" thicker after installation. Is this normal or am I doing
something wrong? I'm pretty sure I've purchased 1/2" and not 5/8" drywall
but I check it tomorrow.

"Old" drywall was 3/8" . Your 1/2" IS 1/8 thicker.

You can still buy 3/8" rock