Do the words in bold sound OK to native English speakers?
(1) A lot of garments branded by a few well-known companies have been copied illegally.
(2) Logos and phrases trademarked by some well-known companies have been copied illegally.
Do the words in bold sound OK to native English speakers?
(1) A lot of garments branded by a few well-known companies have been copied illegally.
(2) Logos and phrases trademarked by some well-known companies have been copied illegally.
You probably want "trademarked" here. It refers to a specific legal intellectual property protection, while "branding" is a more generic term. Something that is branded is simply marked in some way by its manufacturer (or owner, for example in the case of branding cattle). Something that is trademarked is legally protected by trademark law.
That said, garments themselves can't be trademarked: only the logos and names. Designs and manufacturing methods can be patented though.