My 2.5 yo has some redness under his eyes. Is that the same as those when older people have black under eyes?

Possibly. Redness under the eyes of a small child could be many issues. The most common would be allergies or any other source of eye irritation leading to chronic rubbing. This is different from the black under the eyes of adults. You should have the child evaluated for the cause of the redness/inflammation so it can be properly treated.