Yes, the authentication works the same with the web base. I reckon what you have said about having accounts on exchanges and wallets in phones, it shouldn't be done with modded phones. We'll never know who gets to track the phone through that mod and that's why just a reminder to those who uses cryptomus and other financial apps, don't install it on modded apps/apks to avoid troubles.
Hmm, Honestly I used to test some of the Android applications and projects I'm working on so as know I'm the author there's nothing fishy in the code, also I never recommend anyone to install any useless application whether it's gaming or any other on the smart phone you are using your financial apps and your OTP's.
Anyone who wants to know about the best SIM card OTP device in cheap can PM me haha and tag me here too, I'm here to assist.
(I'm sure personal advertisement is not allowed here haha).
Cryptomus I hope this won't fall under personal promotion

.
For the rest Keep Updating us with new features and developments.