Sort of. I'm notoriously bad with names, whether it's names of people, places, music, or movies, I just always have trouble remembering what something was called. So although I'll always recognize an anime I've seen, and may try to sort of pseudo-explain what an anime is to prove I've seen it, I don't have a written-down list of everything I've seen.
For the most part, the only ones I DO remember the names of more often than not are ones I'm presently watching, because..well, if I didn't know what they were called, I wouldn't be able to search for them!