If we use the same principles that are applied to calling your self white and switch them to call someone black you will see how ridiculous it is.
Here is a Old Picture of a African Bantu Tribe
And here is a remote Philippine tribe of the Negrito people
Both have the same skin color hair type and eye color, but both are genetically completely different people and are not genetically related what so ever.
If they where white skinned some of you would say that they are the same race, even though they are genetically different.
So what i'm getting at here is Just cause you have a similar skin tone with someone doesnt mean your the same.
The "White" term was invented in the US when the original colonizers (Anglo Saxons) after many hundreds of years became broadly a mixed people due to immigration. Some where along the line people stopped identifying as British Colonists and started interbreeding with migrants from all around the world.
This mixing caused the majority of the population to lose there cultural roots and so if someone was asked what there background was they couldnt say because they where so mixed that they didn't know.
For example instead of someone just being of Irish descent they may have been Irish/Dutch/French/Spanish/italian.
Basically a bastard race was formed in the US and people couldn't any longer identify as being any of there mixed races so they would just say they are white.