Oh, I give you St. Denis, the church/abbey in France that serves as the burial place for French kings.
Their tombs are the kind that are above ground, with a statue of the dead person lying on top, as if they were on a bed. When you see them all together, you can see how it took the artists some time to work through the issues of sculpting dead guys in bed. (FYI, the bodies are not in the tombs any more, they were vandalized during the Revolution).
So here’s Charles Martel, and you see see that one issue that is clearly, his feet are sticky uppy, but when you have a statue, the sticking up parts are the ones that get knocked off most easily. So, the artist has left a little bridge for his feet to keep everything intact. This is probably also related the the belief that these statues were carved standing up, and then lowered on to the lid. So the feet would have been connected to a base as the artist was working. You can also see this in the robes - they make more sense, gravity-wise, if he were standing. It’s been a long time since I was an art history student, but I remember that when you look at them all as a group, you can see how the various sculptors were obviously trying different ways of making this look more normal. There’s also a theory (I think a little fringe, but I mention it for trivia) that the statues were intentionally made to be standing up, because when the day of judgment comes, the lids will rise up and then duh, you want the statue to look good.
So then you can see that with other kings (and queens, I should add), like Leon V, someone decided that you could protect the feet in a more decorative way, let’s make it a little lion. It’s an opportunity to get a little symbolism in there – lions are kingly, dogs are loyal … I think there’s one princess that has a little fawn, which is sweet.
Your link for Sir John specifies that it came from his burial monument. You see this a lot – there’s a practical reason that the kings have an animal, but then if you are not a king and your monument is not quite so fancy, maybe it is more of a bas-relief picture on top, you still want it to have that “royal” look and they have a dog so you would like a dog as well. Now it’s a fashion that no longer has a practical purpose.