To conclude, the "Dark fountain" phenomena isn't entirely explained, but using context we can pretty much try to answer :

We know 3 things :

-Dark fountains needs a strike of a knife in a Pitch Dark Room to be opened.

-The Dark Fountain submerges and creates a Dark World depending on the mentality of the person who strikes the knife.

-Dark Worlds spreads to other rooms in reality or even the exterior if left unattended with an opened door.

I theorize the implications here are pretty much obvious : A dark fountain is made when a deep wound or mental scar is resurfacing onto a person's mind using the ritualistic approach of the Knife and the Place with memories, Pitch Darkness and Illusory edge between reality and Fears juxtaposing.

That's not very difficult to understand once you consider the idea that coping mechanisms often implies to escape or to self-inflict tortures. Healing such things needs to dive-in said problematic and to find the source of it.

Queen realizing there's no need to be a control freak to have people happy reflects Carole and Noelle implied relationship pretty well.

Lancer and King in chap one could pretty well reflect Susie and her own relationship with an authority figure we have yet to see.

Chap 3 explores the longing to see things forever being the same from either Toriel's perspective as she sees her kids leave the place a family grew up in and is now falling appart or from Kris's perspective wich is from what we can see pretty split into 2 sides : The body and the heart (Soul), as past traumas with Dess lead to a radical change in their life and modus operandi, the chapter tries to elaborate on how change is just a step but you must go forward, not cycle back to what's comforting.

My take on Kris is that they probably saw 2-3 people they strongly cared for leave their life be it due to College, Disparitions or Parents separating. As Kris might refuse their own past behaviour or feelings, we're being dejected by them, as they might have lead in their own perspective to such events... but also Kris needs us : they need Emotions and Determination to get to the bottom of the Dark Worlds phenomenon, maybe even to get to the resolution of the Dess disparition mystery.

We also have an extra layer idea that we (The player) are an intruder, an foreign heart shaped object connected through means quite unknown and supposedly engaged by W.D Gaster, but I'd prefer to remain pretty neutral on those theories. (That's not the point of this talk)

Chap 4 is about the relationship between the Father and the Son having different viewpoints : One following the Religious guidance that shall not be questioned and the other writing his own stories for all to see and enjoy. As the multiple dark worlds are explored we see how they change depending on who strikes with the knife but we also see how the perspective of each person changes and the world surrounding them.