However, someone has told me that if in both lenses I will use f/1.8 aperture, still the latter lens will have more light.

It sounds illogical because it's wrong. At f/1.8, both lenses will let in (approximately) the same amount of light.

I say "approximately" because the f number is derived only from the ratio of the focal length to the size of the aperture. In the real world, other factors, like the amount of light absorbed by the glass also play in — for cases where it's important to know precisely, there's a thing called "t-stop". That's primarily useful in video, where you might need to keep exposure perfectly consistent when switching lenses. In still photography, it's usually just ignored.

Most importantly, the t-stop isn't related to the maximum (widest) aperture of the lens — that is, it's not at all guaranteed that the lens with the widest maximum aperture will have the highest transmission. In fact, since such lenses often have more glass in order to compensate for artifacts at wider apertures, the transmission might even be less.

So, to answer the basic question: the f-stop — the size of the aperture — is the primary factor that makes a lens brighter or not, and different lenses at the same f-stop should give very close to the same results. Transmission is the other major factor, but outside of special cases, it's not considered a significant one. It isn't even generally reported in lens reviews (although you can find measurement of it at DxOMark, which specializes in numeric analysis of measurable aspects of lenses and sensors).