I can only speak for Western Allies... but sometimes. In theory bomber crews had their planes, their crews, their commander and all flew together as a set. In practice, you flew when you were scheduled, in what equipment was available for the mission. Combat casualties (wounded and dead) were high; and the toll on equipment of regular operations was also high. Even without damage the wear and tear on equipment under wartime operation schedules was high. So it could be rare for a crew to fly in “their” plane.

Take for example the plane that dropped the Nagasaki bomb. When it arrived on Tinian it was assigned to Captain Frederick Bock, who named it Bockscar. It flew 13 practice runs, and prior to the Nagasaki bombing, it participated in three bombing missions to Japan. Two of those were flown by Bock and his crew. One of those was flown by First Lieutenant Charles Albury. When it came time to use the Fat Man, the mission fell to Major Charles Sweeney. His plane the Great Artiste was outfitted with an instrument package; it was the plane that recorded both atomic attacks. So Major Sweeney and his crew took Bockscar instead. That was very normal. You had a plane. When reasonable you used your plane. But in war it wasn’t necessarily reasonable very often.

In Enola Gay’s case the commander and pilot of of the plane, Colonel Paul Warfield Tibbets Jr was also the commander of 509th Composite Group and the 1,800 men responsible for the actual atomic missions. He named the plane after his mother, Enola Gay [Haggard] Tibbets. As the commander’s plane it was prioritized for maintenance. And when Tibbets selected himself to fly the Hiroshima mission and his own plane, the ground crews made sure that the commander’s plane would be ready to fly. Interestingly it also flew as part of the Nagasaki mission. Captain George Marquardt flew it ahead of Bockscar, to scout the weather conditions over the primary target Kokura.

So to sum up for your questions. Planes were assigned to a particular pilot and crew, but were flown by other crews all the time. And Enola Gay was the name of Colonel Tibbets’ mom, and was named after her.