Fred Sturm is absolutely right. The historical literature leaves no room for doubt that equal temperament was universally prescribed in 19th century German literature on piano and organ tuning, music theory, composition, and acoustics. ET had already essential become the recommended temperament from about 1750 onwards, although undoubtedly some continued to prefer slight differences in key color. However, as the century draws to a close, references to such a preference become increasingly rare, and ET is ever more roundly praised as being the most perfect of all temperaments. That's not including Kirnberger, of course, whose crude Just/Pythagorean mix was so far outside of the mainstream of prevailing theory and practice that it cannot be considered anything other than an historical curiosity, which as Marpurg said, was "praised by many, used by no one".
In terms of actual practice, it's quite likely that many musicians/tuners continued to use slightly unequal circulating temperaments up through the first decades of the 19th century, either by choice or by accident, but by the times of Brahms birth, ET would have been expected of any competent tuner, even if were imperfectly implemented. Even with Sorge's method (c. 1750), if you are careful and demanding, you can get so close that any deviation is musically inconsequential, especially if you use a well-constructed monochord as guidance, as he recommended. Any technical limitation in terms of implementation disappears completely with the publication of Scheibler's method in 1834. With the various further explications and elaborations published soon thereafter, most notably those of Loehr (1836) and Töpfer (1842), it became quite easy to tune an absolutely-perfect ET, even without the aide of Scheibler's forks. His methodology was referred to and highly-praised over and over again in all manner of subsequent publications, such as articles in the AMZ, books on acoustics, piano making, etc, so any German tuner during the second half of the 19th c. who didn't know about it would have simply been incompetent.
All, Pitch considerations are of significant import in this entire discussion, perhaps more than ET vs. UT vs. WT vs. MT vs. IT themselves. Maybe we should remonstrate against the notion of establishing an absolute standard of pitch much as an absolute standard of tuning. Gerry Johnston observes: "My understanding of history is that E.T. was arrived at in order to 'even out' the various keys by making them all, essentially, equal (as desired by musicians of the time). Any other temperament must have, by the nature of things, unequal intervals. So that the key of "C" with have different characteristics than the key of "D" for example." Actually, this discussion may have neglected to unequivocally observe that this may or may not have anything to do with non-equal temperament. An elusive interview of the late great Earl Wilde-though his pedagogical history and lineage leads to Liszt, not Brahms-can be retrieved by googling "Earl Wilde Interview". Click on the link to the you tube video "Earl Wilde Interview - IKIF 2005 - Part 1 of 3 - You Tube" and scroll down to part 2 and 3 of the same under you tube "Suggestions." He had a reputation for telling dirty jokes; prepare to be offended. At the end of part 2 and the beginning of part 3 he talks about his habit of transposing historical piano literature into lower keys due to the propensity of pianists and piano technicians to demand 440 and above in the 21st century. Actually, he blamed the Boston Symphony, and string players. One of the things about ETD's that became apparent when I started experimenting with them is that pitch itself is transformed into a gargantuan obligation of the piano technician as opposed to just using a fork; it is the inclination of those using a fork to think pitch is less important than it is, those using an ETD, that pitch is more important than it is. The trade-off is that you are more likely to lose a client in a region like this, where the most reputable technicians swear by ETD's, for not tuning at pitch, than not tuning ET. Have ETD's well served us by spoiling musicians so much that instrumentalists are incapable of adjusting to pitch, particularly when considering the destabilizing aspect of pitch adjustment which does not serve our clients well, perhaps, more so, within a reasonable range? The plate certainly reinforced the structure of the modern piano. Period instruments are considerably more sensitive to the tension created by tuning sharper.
Another suggested you tube search: "Brahms speaking." He did record on the Edison wax cylinder. Perhaps what remains to be determined-assuming like with more modern recording devices, the wax cylinder rises in pitch when driven faster, lowers, when slower-is the correct speed that the cylinder should be run, and an evaluation of pitch, not temperament, develop from the recording. Respectfully,