You can argue that the different keys of equal temperament don't have any special character at all, since they differ only by fundamental frequency and the relationship between scale degrees are exactly the same in each key.
What character they do have may come from how accessible they are on different instruments and the association we have with music written on those instruments. C and G major are easy on the guitar for example.
I was a professional harpsichord tuner for a few years in college and composed a few things. Modern brass instruments still do not play perfectly in equal temperament in their low range, although good players can correct most of the way with their embouchure. The fifth is only about 2 cents (2% of a half step) off, which is easily correctable, but the thirds are strikingly different - about 30 cents off - which can sound jarring. I think this is the only class of instruments left in this situation (and it only happens rarely) due to the fundamental physics of the instrument.
Also, certain instruments sound different in different registers, so transposing a concerto written for B flat clarinet up a 3rd (for example) might sound very weird if you played it on the same B flat clarinet. That is particularly noticeable on woodwinds, though, which have very distinctive register differences - most people probably wouldn't be suspicious if you transposed a piano or string piece.
Up to the mid 1900s, pianos were actually tuned to play well with brass instruments that needed the preservation of a particular harmonic series - keeping pure fifths between A flat and C on the circle of fifths, for instance, and heavily weighting the remaining pure fifths in the tuning system toward the flat side of the circle of fifths. A popular temperament of the time was named after a guy named "Kellner," who wrote about it in the 1970's I think. Equal tempering is a very recent invention.
In contrast, the baroque temperaments, like Werckmeister and the theoretical renditions of Bach's "well-tempering" have more pure fifths in the sharp side of the circle of fifths, driven by the fact that string instruments would be tuned with pure fifths rather than equal tempered fifths.
I think this change in tuning ideas is why a lot of the saddest and most emotionally profound music of the romantic period is written with 2-4 flats in the key signature, while baroque sadness was often in D minor, A minor, or E minor.
"The two figures frequently credited with the achievement of exact calculation of equal temperament are Zhu Zaiyu (also romanized as Chu-Tsaiyu. Chinese: 朱載堉) in 1584 and Simon Stevin in 1585."
Yes, people had worked out the pitches hundreds of years ago, but widespread practical adoption really took until the invention of the electric tuner. Tuning an equally-tempered fifth by ear is difficult and unnatural compared to using a slightly unequal temperament.
Great point about the timbres sounding different in transposition. I didn't realise the thirds were so off in brass instruments, thirty cents is a lot!
What character they do have may come from how accessible they are on different instruments and the association we have with music written on those instruments. C and G major are easy on the guitar for example.