My headcanon is that Monsters have organs like Humans. They physically exist, but are made of magic. In some cases (like Skeletons), we can’t see or touch them. But they’re here and can be affected by sickness. Monsters can then be contagious to Humans, and vice versa.
If there’s a need to see or touch the organs (to heal, procreate…), I also use the headcanon of ecto organs or whatever how you call it XD Like, they can make them appear at will. If they're unconscious or unable to do it, the doc uses science to do so. And I like to imagine that checking the soul helps them see where the problem is in case of sickness or wound. Like in “HoneyMustard” for example, Swap!Papyrus has a hole in his soul, its location corresponding to the lungs. I can go far sometimes, ik XD
About procreation, there are other ways I imagined to make my stories possible.
First, Sons in “Pacific Rundertale” and “Frenchtale”, who is my first Undertale fanchild (child character of other characters, created by a fan), and a double clone of Sans and Frisk. Simply because I didn’t consider organs existing in Skeletons back then, and it was the first credible thing that came to my mind XD
Then Nemu in “HoneyMustard”. I just wanted a fanchild for Swap!Papyrus and Fell!Sans, and had to find out something since they’re both males. So I imagined that when Monsters are deeply in love, cuddling (without sexual act) and really connected and everything, their souls each one gives a part to fuse with the other soul’s part, creating a new living being. I didn’t think of a contraception for that case tho. There Fell!Sans wasn’t ready, then Swap!Papyrus got sick, and they now have a sudden huge responsibility. Good luck to them =,D And now that it’s done, I can’t say that the future parents have to wish to have a child for it to work. I mean, it’s possible, just not in that specific AU since I didn’t do it like that lol. Unless they both unconsciously wished for one~
Need some aspirin, or was I clear enough? XD Kudos to you for staying focused ‘til the end!