The issue is AOSP Messaging doesn’t support the relatively new Android system API for messaging. It supports only the legacy API. New Republic 3.0 phones require use of a messaging app that supports the system API. Messaging apps using the legacy API generally work for SMS (standard text) but not MMS (picture and group messaging). They may or may not show text messages as coming from the right number. For better or worse, Republic has standardized support on Android Messages (formerly known as Google Messenger). Please note the name change as that’s indicative of the direction Google is going in. Over time AOSP Messaging will be phased out in favor of Android Messages as the bundled messaging app on new Android phones. It’s already the default on new Moto series phones, the Nexus 5X and 6P and Google’s Pixels.
Depending on which phone you have, you may or may not be able to turn off the Republic app’s nag message about this. If available, the setting will be at Republic app -> Gear icon upper right -> Advanced settings -> Show incompatible messenger. If it’s not there on your phone, the only other option would be turning off all notifications for the Republic app.
Lastly, though Republic has standardized support on Android Messages, it’s hardly the only messaging app using the Android system API. I’ve used both Signal and Textra. Additionally, I’ve tested Chomp, Evolve and Handcent. All work without Republic nagging about their being incompatible. Perhaps, you’ll like one of them better than Android Messages.