Welcome to Android Central forums. Looking at your screenshots they don't appear to start at the same place. What I mean is they don't both start at January and then we see one ends in July and the other ends in June. I did noticed that the button at the bottom disappeared. Out of curiosity, did you try scrolling from the bottom edge, like using gestures, to see if it reappeared? I ask because the way you describe it sounds a little like the button may disappear to give an unobstructed view, and then might come back when you pull up from the bottom to bring it back when you need or plan to use it. In a sense I'm picturing it getting out of your way so you can see more and because you aren't pulling up from low enough on the screen it seems to not return. If you're scrolling in the middle of the screen, everything moves up and down just fine, but if you don't go low on the screen it doesn't bring the hidden option back, which you then mistakenly assume is a cropping thing instead of an animation.
Other things to try would be changing your screen resolution, or display size. Often the screen is set on the lower resolution out of the box to "preserve battery life", and can be changed up or even down which can affect aspect ratio. Display size is also often in the middle OOTB, not wanting to be too small for users with visual issues or old people eyes like me, and not wanting to be too big for normal users with normal vision. My main argument against this is your statement that things start off OK and then shift. If the aspect ratio is messed up, it should always be messed up. One would also think that it wouldn't just cut off the bottom but stretch other things on screen causing them to enlarge, but looking at your screenshots the text appears to be the same size in both images, which is why if you had started both off on the same month it would have been more helpful in seeing the change and difference.
Anyway, what about games, does this same thing happen in game? What apps does it happen in?
What about landscape mode? Do you see this when the phone is sideways?
Let us know what you find and we can offer more advice from there.