From apps like WhatsApp, SMS, Telegram, etc what most of our social contacts use is Signal the best option. Downside mentioned by many about the fact when register a mobile number is needed for the code to confirm send by SMS can bypass with a anonymous prepaid number in a second phone where you can receive the code.
So the Signal app use now on your device isn't with the number from this device. But all your social contacts have your name in their contact list. So not so anonymous after all, but for use with social contacts like friends and family the best option. Even the chats are just everyday, family and normal social stuff, it is yours and should be private. Nothing to hide is not the same as nothing to loose.
For non-social, sensitive, political use, I should go look elsewhere. This way you not mixed up between social contacting privately others and sensitive secure contacting others what if be exposed could harm you.