According to tests both SQL give the same result.
alter table public.company add constraint company_name_key unique ("name")
vs.
alter table public.company add unique ("name");
PostgreSQL (ver. 12.16)
The standard ISO SQL language admit those two syntaxes. In the first one you choose the name of the contraint while the second one will generate a name for the constraint by an internal RDBMS process.
When altering/droping a constraint you need to give the name of the constraint. It is why I will advocate the first one.