How do you define pi * e, with addition? How do you define 2m * 2m, with addition?
Those feel completely intuitive now, but students can get the first far more easily than the second.
Once you approximate them with rationals you can also imagine adding a fraction of the second multiplicand.
Even more basic is the 2.7 * 3.1 = 27 * 31 and what do I do with the decimal place question. Kids first intuition is often 83.7 because it was one from the right in the numbers they started with.
In that context pi * e exposes several different challenges to peoples mental models of multiplication. Granted most people are just going to plug it into a calculator and trust the answer without much thought, but such is life.
It's clear the multiplication as repeated addition holds for the natural numbers, which form a closed ring anyway. Pi and E are sufficiently advanced that by the time you get there you must understand that the operation being described by multiplication is not the same operation at all (depending on how one constructs the real numbers, multiplication of reals is multiplication of sets or functions)