Is the fact proper names are somewhat fixed in language (often not translated and also do not tend to change even in light of a change in attributes from which it gained its name), mean that it is no longer a question of linguistics? Rather confused by its status.
Proper names are nouns, albeit with specific characteristics, so they play the roles other nouns play in syntax (subject, object, etc). They also partake in morphology; if a language has genders, it will distinguish between Cláudio and Cláudia just like it distinguishes between gato and gata; if a language has cases, it will have Cladius and Claudii. The etymology of proper names can be studied just like that of other words, and it is subject to the usual phonetic transformation (Charles vs Carlos); proper names can also be borrowed from language to language. They can even be productive in terms of derivation: Newtonian, Trotskyist, Spoonerism, Franco-Germanic.
So I guess they are a quite proper object of Linguistics.
Onomastics¹ has a big overlap with linguistics, specially with historical linguistics (derivation and meaning of names, sound shifts, etc.). Some valuable historical linguistic information is only available by the study of proper names (several languages are otherwise completely undocumented). It is traditionally seen as a branch of linguistics.
However, there are non-linguistic aspects of onomastics: The study of historical professions (giving raise to modern surnames in Europe), cultural and religious practices reflected in proper names, taboos.
The synthesis of all those fields makes onomastics a fascinating discipline, but there are very few posts for researchers in onomastics out there.
¹ I link to German language wikipedia because it is much more informative than the English version
Proper names are created in many languages, often by taking an interpretable phrase and using it (e.g. the Shona names Chipo "gift", Farai "be happy (pl)!"), the name of the Tanzanian author Kezilahabi "came from a bad place"). They frequently have special morphology (different inflectional affixes; only they take the "honorific" prefix; often wider use of and special forms of diminutive affixation) and syntax (especially obligatory vs. prohibited definiteness marking). Even the study of why people name their child "death approaches" is within the sphere of one form of linguistics, sociolinguistics. So I can't see what wouldn't be linguistic about the study of proper names .