Lenses described as "warm" tend to allow longer (red, orange, and yellow) wavelengths through at slightly higher rates of transmission than shorter (violet, blue, and green) wavelengths. Lenses described as "cool" are the opposite, they let a little more of the shorter wavelengths (violet, blue, and green) through than longer wavelengths (yellow, orange and red) when equal intensities of the full visible spectrum are falling on the front of the lens.
This was more of a concern in the days of chemical film when a photographer was limited to the emulsions available with various films. Even with panchromatic B&W film, a 'warm' lens will allow red objects in the scene to be lighter shades of gray in a B&W print than equally bright blue objects in the scene. On cloudy days 'cool' lenses tended to give the sky a slightly purple/magenta tint with daylight balanced color film. On cloudless days a 'warm' lens would give the sky a slightly yellow-orange tint with the same daylight film emulsion.
With the ability to individually process raw digital image files this is much less of an issue than it was with chemical films. Digital photo processing allows the equivalent of designing a different custom film emulsion for each and every individual raw file we process and convert - and we get to do it after the fact! Not only can we control color temperature and white balance, which move pretty much all of the colors in a particular interpretation of the raw image data in one direction or the other (blue ←→ amber for color temperature, green ←→ magenta for WB correction), but HSL/HSV/HSB tools allow us to adjust the hue, saturation, and luminance/value/brightness of multiple different wedges of the color wheel independently of the others.
When Photophiles speak of the "character' of a lens, they're usually more concerned with how a lens renders a scene in terms of resolution, especially with regard to how sharp the in-focus areas of the scene are and how the out-of-focus areas are blurred. Things considered are the shape of blur as well as whether the blur has a uniform brightness, is brighter near the center or the edge, or even has sets of brighter and darker rings between the center and edge. A "smooth" lens renders the out of focus areas in a nice creamy way with the details of the background (or foreground) melted into a homogenous blob with smooth transitions from areas of one color and/or brightness to another, while a "busy" or "harsh" lens will render the details of out-of-focus areas in ways that can be distracting to the viewer. The direction and amount of field curvature and spherical aberration (two of the seven classic optical aberrations) significantly affect the "character" of a lens and the way out-of-focus areas of the scene are rendered as well as influence the lens' absolute resolution at the focus distance.
Even a lens made perfectly according to its "blueprint" has optical aberrations that are caused by the real thickness of refractive materials. The only theoretically "perfect" lens with no aberrations would be a theoretical lens with zero thickness. Most of the classic optical aberrations are due to the the fact that different wavelengths of light are refracted at slightly different angles at the air/lens boundary on the front and back of lens elements, or at the boundary between two lens elements glued together that have different refractive indexes.
How lens designers decide to deal with optical aberrations and how much correction they design into a lens by using various different elements made of various refractive materials affect the "character" of the lens.
What refractive materials are used in each of a compound lens' elements as well as what anti-reflective coatings lens designers choose to use to reduce lens flare and ghosting can affect the color transmission profile of the lens, which is what is often referred to as "cool" or "warm".