Except in children, fat cells don’t generally increase greatly in number (“pop into existence”) during weight gain. Existing fat cells, however, accumulate more fat within their interiors. Adipocytes (fat cells) are everywhere throughout the body, along with their support cells to supply structural scaffolding and blood supply.
They all carry the same genome, producing fat under the same genetic control mechanisms (not fully understood) inside the cell. While scientists are still working out the various stimulatory & inhibitory hormones & other cell signals, within any single individual the levels of these signals should be constant, because of thorough mixing in the blood. Hence to some extent all body fat increases or decreases according to dietary calorie excess or deficit.
There is a “higher-level” genetic program that determines the distribution of fat in various regions of the body in both normal and obese states. For instance, women accumulate more fat in the hips and buttocks while men accumulate more fat in the central abdomen. Somehow the fat tissue “knows” which region of the body it’s located in, and varies its sensitivity to fat regulation accordingly.