"Character" is one of those loaded English words that can drive people learning the language crazy. Big difference between
"He is a character" and
"He has character." or,
He plays a character." One means idiosyncratic, the other moral or ethical strength, and yet the other means to mimic.
When regarding photographic optics, my definition of
character is the root meaning:
the combination of qualities that distinguishes one thing from another. Otherwise, we go off into the weeds as far as communication is concerned.
IMO, all lenses have character, even $6,000. ones, and that character need not be "flaws" or Holga like performance, nor retina slicing sharpness alone
The Contax 120 APO Macro has a character that visually separates it from other 120 macros, but is hardly loaded with flaws. So, what those image characteristics are, and what appeal they may have is all in the eye of the beholder, and what the photographer wants to achieve with his/her imagery.
In context to the OPs original question, quoting another thread where I forwarded the notion that modern MFD lenses have become less defined by "sharpness" ... my personal feeling is that the focus on that "sharpness" and "resolution" has reached obsessive-compulsive levels to the detriment of other characteristics that appeal to the human eye on emotional levels.
Subjectively, I see a lot of sterile and lifeless 2D looking images lauded for their sharpness and resolving power ... with 300%+ crops to prove it. However, the over-all image is flat as a pancake and subjects often look cut out and pasted on their backgrounds ... including some from $6,000. optics.
As long as this obsession prevails, the lens designers will full-fill it as users vote with their wallets. I also believe that, for the most part, this forum is highly engaged in this pursuit ... but not all photographers are shot in the butte with the resulting lack of other characteristics ... and I am one of them.
-Marc