"Pushing the envelope" was very meaningful in its original context which was high-speed (faster than sound) flight.
The term predates the time when "faster than sound" was nowhere near the "envelope," although it does originate with "flight."
Every airplane has a specifically defined "flight envelope," that describes the range of flight conditions within which safe operation is reasonably certain. There are a wide number of "standard limits," such as Stall speed (Vstall), minimum Takeoff speed (VTO, Maximum Flight Speed (VNE also called Vcome-apart, and for most planes Maximum Bank Angle (based on g-force limits and also fuel system limitations).
The "Envelope" has been a recognized set of parameters specific to each individual airplane since at least the early 1920s.
"Pushing the envelope" has been understood, by pilots at least, to mean "operating in an unapproved manner" since then. There was a (fairly brief) period when the "envelope" was close to Mach 1, so that in that brief time the envelope might have been said to have some relation with that speed, but that time was just a flickering instant.
It's an unfortunate "fact of life" (add that one to the list) that test pilots sometimes must push the envelope in order to explore just where the actual limits are before the customers get the plane. There are long lists of "heroic but dead pilots" who did their duty in this way.
As adopted by management, "pushing the envelope" just means "lets do something I never thought of before" (i.e perform a miracle that will save my ass) - an admission of ignorance, and expressing NO CONCEPT of risk, since the manager never dies - there's always some flunky who can "pay the consequences" (be blamed) if something goes wrong.
A similar phrase - "let's expand the envelope" - might in some cases have some meaning, but it's rarely heard since it might actually involve improving the product to make it do more (which might cost money).