Board Thread:General Discussion/@comment-29855659-20170630205901/@comment-13682699-20170630225600

Way back during the War, Humans could use magic and they also had a way to combine their power (such as when making the Barrier) so it is relatively unknown how their power worked at the time. Its highly probable that only one human (the one with the highest determination of all of them) COULD reset, BUT because all the humans had similar levels of determination, they would remember it as well and use that against the monsters so that there were no casualties.

Flowey remembers your resets because he had enough determination at one point to reset himself. Even the previous humans that fell in the Underground had a limited ability to save/load/reset as mentioned by Toriel when she says that feels like she's met you already (after meeting her again from a load/reset) and felt the same way with the other humans as well.