Hellcat was the name of both an American carrier-borne fighter aircraft, and an American tank destroyer. Firefly was the name of both a British carrier-borne fighter aircraft, and a British (mainly) tank destroyer. Almost all of them had their combat debut in 1944 (only the Grumman F6F Hellcat already in September 1943). Is this entirely coincidential, or has there been any influence in any direction? Like, some Englishman said: "Hey, they Americans called their new tank destroyer design like one of their naval fighters. Let's do the same! What's our most current type? The Fairey Firefly? So let's call our new Sherman-based anti-tank design Firefly!"
Actually not. The USN and USAAF officially began attributing names to aircraft long before the U.S. Army did for tanks and other AFV. It was May 1944 before the Ordnance Department began recommending names for Ordnance fighting vehicles, the request was renewed in November 1944, but it was postwar before the Army actually officially recognized the naming conventions we now think of for these vehicles. However, the 76mm GMC M18 was not even given its name by the Army, it was given it by Buick, who built them, in wartime advertising. Firefly was also not a name for the British-developed adaptation of the Medium Tank M4. It was actually the code name for a series of projects for a self-propelled version of the 17-pdr antitank gun developed as a possible substitute for the Challenger 17-pdr. It was not a term even frequently used for the vehicle during the war, instead, most references are to "Sherman 17-pdr" or "Sherman Ic" or "Sherman Vc", which was the actual official designation.
Yes, after contracting for them the British named the Medium Tank M3 as the "Lee" and "Grant", depending on turret used, and the Light Tank M3 as the "Stuart". After that it got complicated as various characters added and then subtracted "General" to the titles, but the U.S. Army never officially recognized them for its own use.
Firefly is such a common term used for that British upgrade for Sherman---I have a feeling that many British tank crews must have called it that at the time. it might not have been an official term, but it must have been used fairly commonly at the front.
They may well have, but when they began using it commonly is impossible to say. The first actual references I know of to it by troopers are Desmond Fitzgerald's History of the Irish Guards in the Second World War in 1949 and in Roger Evans', Story of the Fifth Royal Inniskilling Dragoon Guards in 1951. Then B.H. Liddell Hart used it in volume II of The Tanks in 1959, and Steel Brownlie used it in The Proud Trooper in 1964. It does not become common in print until the 1970's. In wartime documentation it is hardly used at all in the context that is now familiar. It does appear in a few war diaries, which does indicate it was in use, but whether or not it was "commonly used" remains the question.