px_eliezer_ Vaccines do have brand names...
For example, in Canada there are about 10 approved Flu vaccines, including Fluad, Fluzone, Afluria, and others.
The public just never got used to thinking about the Brand names of Covid vaccines, it was more the manufacturer name (Pfizer–BioNTech, Moderna, and the various others).
It all stems from the unprecedented and chaotic nature of the whole pandemic rollout.
The issue was that they couldn't really market them until they received full approval. So, while under emergency use they had to just use the company name.
A lot of medications tend to have bizarre names for trademark purposes. You want to avoid confusion and you need a name that isn't descriptive. So the marketing team needs to come up with something memorable, that also meets trademark requirements.
It's always kind of strange to me how much thought marketing teams put into the meanings behind company names and logos. Super subtle things that I think most people aren't even aware of. Sometimes all the effort that goes into explaining all those different aspects of the name/logo feels like someone is desperately trying to justify their job.