Generally for the Navy and Marine Corps, a pilot was assigned a specific airplane. Most of the thinking on making sure each pilot had a plane with a specific side number goes back to squadron organization. For example, in a pre-war F4F squadron, the CO would fly x-F-1; his wingmen would be in x-F-2 and x-F-3. The exec would be assigned x-F-10 and his wingmen would be x-F-11 and x-F-12. The FO would have x-F-16 with wingmen x-F-17 and x-F-18. The EO would have x-F-7 with wingmen x-F-8 and x-F-9. GO was x-F-4 with x-F-5 and x-F-6. Last but not least, MO would have x-F-13 with wingmen x-F-14 and x-F-15. Knowing how many planes were in a squadron, one could determine the pecking order by reading off the side numbers.
When the squadrons started getting larger in the late spring / early summer of 1942, going up to 27 and then to 36, much of the logic behind who got which plane went out the window and what remained was the CO usually getting the -1 side number and the exec getting the side number closest to the middle on the complement.
With the increased tempo of wartime operations, there was even less opportunity for a pilot to fly in the plane he was nominally assigned. For example, when VF-3 went out on Yorktown for the Midway job, there were 27 planes in the complement. Two of these were wrecked in a flight deck accident as the ship was departing, #13 - the exec, Don Lovelace's, plane and #14, his wingman, Evans. Lovelace was killed and both planes were stricken below for depot level repairs upon return to Pearl. With his exec, and close friend, killed, Thach turned to the next senior pilot, the FO, a LTJG from the VF-42 augmentation and appointed him as exec. Pederson, the CAG and previously the CO of VF-42, endorsed his one time, pre-war, wingman as the new VF-3 exec. The fighting squadron maintenance troops were not from VF-3; they were all assigned to VF-42 and had been aboard the ship since June 1941. When they heard that one of "their" pilots was moving up to exec, they struck his plane below, painted over the existing #26 and replaced it with a #13 so that he would have the "correct" side number; and his wingman, another VF-42 type, got the same treatment with his #27 side number replaced with #14.
But even pre-war, aboard a carrier, such assignments were largely paperwork exercises, you flew what was spotted for your use. While a CO might have his assigned plane rolled into his place in the spot, everyone else took what they got.
Wartime tempos meant that even a squadron CO might not even get his plane in the right place in the deck spot. Again, using VF-3 at Midway; Thach, the CO, flew #23 in the morning strike escort mission and later, as the Japanese torpedo planes were starting their run in, took off in his assigned plane #1 to meet that threat. #1 had not, up until then, been used that day. In fact, of the 25 VF-3/VF-42 pilots who flew from Yorktown that day only 4 actually flew the plane to which they were nominally assigned
Ashore, whether or not one flew one's assigned airplane could be influenced other factors. In the early days of the Solomons, aircraft availability AND pilot availability were the driving factors and niceties such as specifically assigned aircraft pretty much out the window. Later, by the early summer of 1943, things were quite a bit less chaotic, steady streams of replacement squadrons, aircraft and personnel led to a more established routine. For example, VF-11 operated out of Fighter 1 on Guadalcanal from April through July 1942. My father was the FO and was assigned #F-21, bn 11985. He flew that particular aircraft in 46 of 49 missions during that time period. I once asked if that was typical and he said that it was. VF-11 was a somewhat oversized squadron with a few spare aircraft that were shuttled about when regular planes were in maintenance. On the three missions where he was not in 11985 he was in 12080, once on 10 May for a Henderson Field CAP and twice on 19 June, once for a photo recon escort to Munda and back and then a return to Munda for a shipping strike (that was 6.6 hours of flying time that day).
By and large, the less organized chaos surrounding the availability of aircraft the greater the probability that one would fly in one's assigned plane. Aboard a carrier, all was organized chaos, so it would be actually rare for a pilot to fly "his" plane. Ashore, it was much more likely to happen, but was greatly influenced by factors revolving around the tempo of operations and the relative sophistication or primitiveness of the operation.
Rich