Marine regiments are designated and referred to as "Marines" in the same manner army regiments are referred to by arm of service - 1st Artillery, 1st Cavalry, 1st Infantry, etc.
Furthermore, marine brigades and divisions are referred to as such. They're NEVER properly referred to as "Marines" First Marine Division is ALWAYS referred to as First Marine Division (or First Division). Likewise, First Marine Brigade. Only First Marine Regiment may properly be referred to as First Marines.