The problem is... no one here (besides yourself) knows what these kids actually look like... so there's no way to tell if their faces are stretched either way. That said, how would you know if the subjects in the photo thread actually look like you think they do?
The S5 lens is spec'd at 31mm while the S7 is spec'd at 26mm. The S7 definitely has a wider lens so I wouldn't expect both phones to take the same photos from the same distance. Not to mention that you'll see more distortion coming from the S7... especially at the edges. Take a look at the curve in the TV chassis and TV stand leg in your second fan photo... not to mention the slanting of vertical lines of the wall to the left and the bookcase in the last two photos. Typical of wide angle lenses and exactly what I would expect to see.
If you want less distortion in your photos, especially people's faces, then you need to step back, zoom in, and frame your subjects. In the old days of 35mm... the recommendation for portraits were lenses in the 90 - 105mm range. This ensured the subject's face was less distorted horizontally and vertically. Even back then, (and still now) if you shoot a portrait using a wide angle lens, distortion will occur for sure unless you are very careful about distance to subject. And... to alleviate more possible distortion, you need to keep the subject centered in the frame. The reason the last picture of your fan is round and straight, and those round things floating in the pool in the first photo are in fact round, is because they are all framed in the center where the lens has the least amount of distortion. It's the nature of the beast for wide angle lenses that anything outside center frame will be more distorted... leaning, slanted, or stretched (if you want to call it that).
All that being said, yes, the problem is because the S7 has a wide angle lens... for sure wider than your old S5. Whether you have to live with it, the answer is also yes... if you keep the S7. However, you can also learn more about photography and remediate a lot of the problems so you're not so bothered by them.
EDIT... I wanted to state that I'm in no way saying "nothing" is wrong with your camera. Maybe it does distort more than others, but from what I see, it looks like what I would expect. You might take your camera to Bestbuy and do a side to side photo comparison... on the same subject... same focal length... and "exact" same framing, to see how yours compares.