The OP is correct. The concept of "race" is an artificial social construct, and almost entirely modern. There simply is no such thing as a "white race" or a "black race" or a "brown race". That doesn't mean that terms like "white" and "black" aren't helpful at times in discussing social issues, but there is no such thing as discrete "races".
While skin tone and color are real things that exist in humans, these are the result of thousands of years of climate and geographic adaptation by various people groups as humans migrated across the globe. People located nearer to the equator have darker skin, people further away from the equator have lighter skin; because melanin--the pigment that gives skin its hue--serves a good job in protecting the skin from UV light; equatorial regions in the world get on average more sunlight. Conversely, as we move away from the equator toward the poles sunlight becomes far more variable, and less intense; fairer skin (less melanin) helps absorb more UV light which is necessary because vitamin D is produced chemically in the body this way. While each and every last one of us can trace our ancestry to Africa, many of our ancestors migrated out of Africa and slowly settled the rest of the globe, and over generations acclimated. Certain features (such as skin color) reflect adaptation to climate, other features demonstrate a close relationship within certain geographical areas, such as hair texture, the shape of the eyes, etc. These differences are, fundamentally, superficial and really only speak to the sheer diversity of our species as we came to occupy nearly every square mile of land on earth.
So northern Europeans have fair skin, a reflection of their northern location; and also tend to show other melanistic mutations such as blonde hair, red hair, blue eyes, and green eyes; the overall human norm is brown or black hair and brown eyes; different colored hair and eyes arises most common in populations where there has already is a dominant mutant gene that makes skin fairer (the same pigment that makes skin dark is the same pigment in hair and eyes, less of it in the skin means lighter skin, less of it in the hair means red or blonde hair, less of it in the eye means blue or green eyes).
If you travel south to the Mediteranean, the people there tend to be swarthy, darker in complexion, and south of the Sahara people are even darker. The same thing can be seen if you go eastward, people in northern Asia are lighter, when you reach the southern tip of the Indian subcontinent and Sri Lanka people are very dark. And we can see this in eastern Asia and Oceania. The same happens in the Americas.
We don't have "races" based on skin color or any other features. We have a world-wide gradient of human variation, in skin color and other features. People from one location are most like people from a neighboring location, and so on and so forth. Egyptians are going to be similar to Berbers, Arabs, Persians, etc, but a Persian is more likely to be like an Azerbaijani than a Berber. Someone from Mongolia is going to be more like someone from northern China than from southern China, while someone from southern China is going to be more like someone from Laos or Vietnam than someone from the Korean peninsula. It's a gradient, and in the last five hundred years we have seen a slow and rapid increase in migration and immigration all over the world, and so societies are now being more diverse with the increase of immigrant people.
Countries like the US and Canada are melting pot nations, immigrants and the descendants of immigrants outnumber the original native peoples by large margins, and further immigration over the last couple hundred years continue to see an influx of diversity of the population. It is largely in this relatively unique historical situation that the idea of "race" was born, primarily among Europeans in the 18th and 19th centuries who sought to justify slavery, the massacre of indigenous nations, by believing their "whiteness" gave them some kind of claim. And when various suffrage movements, such as the abolition of slavery in the US, suddenly turned former slaves into freemen, it fostered a racism that has continued, slowing down only marginally, to the present day.
Race-based ideas borne out of European imperialism and pseudo-science.
-CryptoLutheran