On my S22 all you have to do is open a chat and press the three dot menu and select Customise chat room this way you can have a different background for all your chats . Note some chats usually no reply numbers you cannot change. Hope this helps.
That's strange are you looking at a message from a personal contact or a replying blocked message usually you cannot change the background on those kind of messages. But I've changed the background of my personal contacts this way on my S22 without any trouble.
The Samsung messages app should be installed on your phone from new have you been using a different message app, check your settings under Apps the Samsung app should be there. I have looked on the Galaxy Store but the Samsung messages app is missing for some reason. If it's already installed on your phone then changing the default messages app in the settings should work .