So yes. Yes is what I am reading. The designations are more like generic "make/model" numbers and not unique "serial" numbers or names. Thanks for the reply!
If C-3PO is a 3PO series protocol droid, I would find it hard to believe that only 26 (36)? were ever made and each one had a unique name.
R2 is an R2 series astromech droid. I would plead a similar case there as well. Especially given the probable number of astromechs built/needed.
Their are many more droids than the easy designation numbers, and droids don't have to follow standard designations at all.
Think of it more like the deragatory terms used on enslaved humans in our own history; the desired names of slaves mattered little to the masters.
Sentient organics are basically saying "yeah, I'll give you one extra letter to make it easier to call you instead of the other 3PO chattel"
C-3PO may have a completely different name they would prefer to self express as; but in a universe where even sentient organics are enslaved, droids don't stand a chance.
To your first point, the droids in question do seem to follow the "easy designations" as you put it. In fact, they are often referred to in the film by them. Examples: Luke often refers to R2-D2 as just R2. And C-3PO as 3PO.
I still don't buy into the notion that these were unique "names" at all and not "model numbers" though, but thanks for the discussion.