So what exactly ARE save points after all? We know that Determination is what let us use saves in Undertale, but was it ever explained what the Save Points themselves were?
Deltarune seems to be making a point of them being something unique to only us:
First Save Point we ever get:
>* At times, you see it flickering.
>* The light only you can see.
>* By second nature, you reach out, and...

The Save Point in the Light World
>* (You could barely see it. A patch of sunlight that shouldn't be there.)
>* (It felt different in your hand. And you knew what it was.)

The Save Point in the big hall of prophecy glass in Third Sanctuary
>* (All around you, blue glass glitters like crystals.)
>* (As you approach it, the glowing light shines through them...)
>* (... creating a prism only you can see.)