Same issue here.
Android notification issues
I have the same issue. I have 2 phones, an old iPhone 5C which has notifications working all the time, and a new Sony Experia XZ3 which only lets me know about 50% of Chess.com notifications. Other apps are fine, just seems to be Chess.com and andriod.
Clearly the app is not liking newer versions of Android. I hardly use the app now and access via the website with Chrome for Android. It's not a perfect experience as there are some bugs even accessing the site with mobile web browsers but I get more functionality than the app by far!
The app just does not have all the features that the website has. For e.g one cannot save one's analyses in the self analysis board in the app, but one can on the website. No puzzle rush on the app. No new format lessons. In the messaging section one cannot insert pictures or a game like one can do on the website. One cannot go back and review game analyses in the app. It does not even save a game analysis! No link to play one of the live computers to a live match if one wants to. It's on the website! Want to play an unrated daily game with takebacks? You won't find the takeback option in the new game setup section of the app. I can go on.
I have also had issues with notifications. The Android app also sometimes generates a notification saying I have only 8 hours left to make my move in a daily game and that time is running out. Meanwhile in reality I have say 2 days to make my move in a 3 days per move game. So a bizzare notification. And I have received that very same notification saying 8 hours left a few times, on different games.
I think the Android app needs a real proper full overhaul to iron out numerous bugs and bring new features that are considered important (like the ones I have mentioned and more) for a great user experience, that is on par with the desktop website experience at present. The app should in an ideal world give users the same tools and options as the website!
Some times when my opponent moves, I don't get a notification, other times I do. Some times it's delayed by hours. It seems to be related to the android app as I have the app on an iPad Air 2 as well and there I get notifications instantly as far as I can tell. I don't use the iPad that often but it has a time stamp on the notifications.
I have a Samung S7 running Android 8.0.0 and I have turned power saving off for chess.com app just to be sure it wasn't the system who was causing this.
Another issue I have is when I start an analysis of a game, I get a pop-up notification and it vibrates for every move it has calculated which is incredible annoying when it does it for 5 minutes. Not to mention that I have turned vibrations off in the app-settings. This still happens when I leave the app.