The way I see it, these seasonal texts are supposed to be taking place in a post-pacifist world. In the game it is implied that Alphys knows the true identity of Flowey (hence why she injected that specific Golden Flower with DETERMINATION to begin with). Before Frisk fell down, Flowey had the power to SAVE and have spent an absurd amount of time doing everything that he could possibly think of by resetting the timelines at will. We do actually get some very specific information from Flowey on some of his previous runs, like how he spent much time with Toriel in the Ruins and Asgore in New Home, but left them in the end because of his lack to feel anything and Asgore refused to let Flowey even near the human SOULS, forcing him to retry in all sorts of ways.

Keep in mind that when we reload our save file, some monsters do show signs of retaining at least some essence of memories of our previous runs.

Which is the part Frisk comes in.. Flowey has never let time advance this far before in his saves, but from what we know of his current run is this:

1. He left New Home immediately after revival, noticed only by Alphys.
2. He befriends Papyrus, telling him things before they happen and encourages him.
3. He spends a lot of time near Chara's grave in the Ruins, maybe even waiting for someone like them to fall down again like he remembers (oblivious to the fact that an determined human would trump his save powers)

Now we spoil forward, Asriel destroys the barrier and retreats to the Ruins before anyone else sees him. Toriel being a boss monster may have sensed the presence of Asriel while her SOUL was absorbed, this in combination with the fact that Flowey actually maintained a lot of Asriel's habits that she of all people would immediately recognize.

So in a moment of weakness, she calls the flower by their true name along with a wishful promise.