Lenses do have different characters- the feeling they evoke in different scenes can be radically different.
In still photography though I feel that most often we regard character as a failing.
I happen to love lens flare, but only in lenses where you have to work to provoke it, like my Hasselblad HC 80mm which gives a gorgeous atmospheric purple haze when a hairlight or the sun is just out of frame and pointing right at camera. A similar scene with a film-era 35mm Canon prime had so much flare that it almost obliterated the subject. Putting a lens hood on the Hasselblad almost eliminates the flare, but also eliminates a certain retro glamour when shooting pretty girls into the light of the setting sun.
So "character" is a lens defect I like, I guess. Of course, the character of the lens starts with its focal length and aperture setting. The feeling of an 85 mm lens at f/1.4 is not the same as a 20 mm lens at f/11! But beyond that, one person's character is probably someone else's unacceptable defect. As someone else said higher up, I like vignetting so much that I not only turn the corrections off in post, but usually slap on a great big 1+ stop vignette on top of whatever the lens delivered. Nonetheless, I can see that vignetting is a flaw.
The game changes completely in movie pictures. A lens that I find almost unusable for stills like a Russian 58mm f/2 that's soft, vignettes, flares and generally looks like rubbish in stills suddenly looks magical when doing a dolly shot through a string of Christmas lights on a backlit subject.
I have an anamorphic adaptor on an elderly Canon stock zoom specifically for blue anamorphic flares, squashed bokeh and the like. Point that at a few practical lights on a darkened set and instantly you get that "Die Hard" action movies vibe. You can enhance it or play it down, but nothing gives you that look and feel like having the right lens in front of the camera.
I think it is also interesting that Cine lenses, which are expensive and "perfect" beyond the dreams of many stills lenses, are too perfect for a lot of uses. You'll hardly find an Arri Master Prime being used without some sort of diffusion in front of the lens (at least not for shooting people). Its like the design has taken so much character out of the lens that you need to put it back with halation, halos around light sources, and the flattering effect on skin tones of black mist diffusion.
All in all, I do have favourite lenses for favourite things, and they are not all created equal. I think there's something close to magical with my Hasselblad HC 80mm, the Sony 85mm f/1.4 GM, the 55mm Russian tank, the wide-angle shallow DoF combo of the 35mm Sony f/1.4 (which I've either got lucky and got a good copy of, or which I find endearing where others find it annoying).
I have similar effective focal length/aperture combinations in other systems, like my old Canon 85mm prime. It was a favourite lens for a few years, but the Sony is better by every metric and that includes how much I like the results. It's like seeing the same character only in high definition, somehow.
I've also had a few lenses which I've just not loved over the years: the 100mm Canon macro from film era was shockingly sharp and unflattering on skin, I never really warmed to the 16-35mm Canon f/2.8 Mark I, and the Panasonic 12-25mm f/2.8 never comes out and about with me- I always choose an older Olympus four-thirds 12-60mm f/2.8-4 instead. Not particularly because of the longer reach, but because when I look back at my favourite shots from the GH4, I took most of them with that lens.
Sure, we can wax lyrical and we're probably imagining half of what we call lens character. But objectively there clearly ARE factors which determine a look (remember the anamorphic flares...) and I think subtle character can lead us to enjoy using one lens much more than another.
Cheers, Hywel