In my childhood days, I used to see everywhere people being condemned for saying "My names are Scofield Oloo," and the condemnation used to make sense because at the end of the day, that is not good English. Or so I thought.
Over the years though, I realised that language is as much a product of culture as it is a vehicle of its propagation, much as blood is a product of the body and the vehicle of its sustenance.
As a result, I came to the following conclusion: "My names are" is as correct as "My name is" for the following reason; English, like most European culture, is as dull as their food. This also includes their naming conventions.
For instance, due to the natural propagation of family names, a bloke from the a line of blacksmiths of Southamptonshire would be called John, therefore his name would be John Smith. Smith is not John's name; it belongs to his family.
On the other hand my people on this side of the Mediterranean have a much more flexible relationship with names. These can change many times over the years for any number of reasons and be added and removed at will.
Therefore, a person's names are theirs. Family-related nomenclature include the names of their clans and certain names that are handed down through specific families or clans which enable a person in the know to trace this person to their bedrooms.
Another reason that establishes the correctness of "My names are" is the abundance of variety of meaning of names. People are given names for various reasons including but not limited to aspiration, uniqueness, circumstance and objects (lack of ideas). A person called Titus Otieno Birika has names that mean different things. "Titus" "Born in the evening" "Kettle" (I am sure Titus has a meaning) is therefore a set of names and not just a name because kettles are not born.
So, dear pedant, my names are "Brian" "Butcher of language" "Something" "Something" and there is nothing you will do about it.
At this stage, I would like to mention that our governments are arbiters of Europeanisms because it is easier for them to work with fixed nomenclature over a person's lifetime than to accommodate their culture.
English has many weird aspects to it, exactly like an unholy amalgamation of retardation would develop over several centuries but without those, flat as it is, it would be flatter than European food, therefore we will not handle those here.