According to Art. 9 CopA, “The author has the exclusive right to their own work and the right to recognition of their authorship”. Two aspects must be taken into account.
Firstly, the author can decide to choose under what name they wish to be recognised as the author. They can use their own name, a pseudonym or even choose to remain anonymous (Philippin E. in: de Werra/Guilliéron (editors), CoRo, Propriété intellectuelle [Intellectual Property] p. 71).
This means an artist who decides to publish a work on social media can choose how they wish to be named as the author of the work in question. Anyone who decides to reuse the work, for example by making it available to other users on social media, must respect the author’s wishes by indicating their name or pseudonym or note that the author is anonymous (Barrelet D./Egloff W., Le nouveau droit d’auteur [The New Copyright], Stämpfli Editions, Bern, 2008, p. 46).
Beyond the actual publication of a work, the right of paternity also allows the author to have their authorship recognised. This means the author can challenge a third party who has usurped the paternity of their work. Art. 25 para. 1 (2) CopA goes even further by forcing the user of the copyrighted work cited to indicate the source as well as the author of the work, otherwise the user risks being accused of plagiarism (Rebetez M., Internet, les réseaux sociaux et le droit d’auteur [Internet, Social Networks and Copyright], p. 56).
The specific way in which social media function leaves users open to breaching the right of paternity. In fact, publishing a work without the right to do so and without mentioning the same of its author is a violation of Art. 9 CopA. Any subsequent users will likely not cite the source of the work in question either, causing a chain of violations.
There is no specific legal framework for social media. This means the author retains their right of paternity of the work and you have to mention the author when they have chosen not to remain anonymous. If the work is referenced via a quotation, you need to do some research to make sure you have quoted the source correctly, in particular when good scientific practice needs to be respected.