We classify historical period based on how saliently we can speak of it as a unit. Thus, as a first approximation, calling Brahms “Romantic” simply means he is like other “(German) Romantic” composers (Beethoven, Schubert, Schumann, Mendelssohn, Liszt, Strauss, Bruckner, etc.) to a greater or lesser degree, whatever “Romantic” may mean. “Romantic” specifies the fact that Brahms comes from a specific time and place.
“Romantic” also specifies the way in which the composers to whom we assign the label are similar. It might well speak to the “effect” or “mood” of the music: these composers are similar insofar as their musics affect us similarly. Or it might be a structural description as well: we might not feel affected in the same way upon listening to Brahms and Beethoven (or Bach and Couperin; or upon looking at paintings by Friedrich and Delacroix) but we might understand them to be exploiting some of the same techniques, albeit to different ends. “Romantic” specifies how Brahms embodies or contradicts the conventions and values of his time.
Finally, the term “Romantic” might speak for itself, as it were. Without knowing about other Romantics, we may have an aesthetic experience when we listen to Brahms (or look at Friedrich) — and by that singular experience alone come to an understanding (at least partially) of Romanticism. “Romantic” is the name given to one’s individual aesthetic experience of Brahms.