This lack of surnames is often a result of the white parentage of many of these slave children, just as is likely the case for Chambers/Tom.  Frederick Douglass, in Narrative of the Life of Frederick Douglass relates this issue.

"The whisper that my master was my father, may or may not be true; and, true or false, it is of but little consequence to my purpose whilst the fact remains, in all its glaring odiousness, that slaveholders have ordained, and by law established, that the children of slave women shall in all cases follow the condition of their mothers..." (311)

"The master...must stand by and see one white son tie up his brother, of but few shades darker complexion than himself..." (311)

"Thousands are ushered into the world...who...owe their existence to white fathers, and those fathers most frequently their own masters." (312)

Harriet Jacobs also relates this reality in her Incidents in the Life of a Slave Girl:

"Always it gave me a pang that my children had no lawful claim to a name.  Their father offered his; but, if I had wished to accept the offer, I dared not while my master lived." (223)