I don't think any of this is really true? If I was Lutheran, I assume I'd just describe myself as "Lutheran".
This whole subthread is pretty weird. There's no deeper meaning to my describing myself as "Catholic". It's how any Catholic would describe themselves.
I think the bit about some "Christians" (mostly American evangelicals) not regarding Catholics as Christians is definitely true.
I encountered such discussions living overseas. For people who come from traditions from outside the Christian world, few know the differences between the various branches or the complex history or rites or terminology, and tend to lump them all in together regardless of the identity or faith of the person they’re speaking with. This is also true of outsiders regarding Islam, Buddhism, or other religions with long and splintered histories.
TBH it’s hard for many people who were raised in a specific Christian faith to concisely explain many of the differences themselves … I would struggle when asked “what’s the difference between Catholicism and Christianity“ or “Catholics and Baptists” back then.