In my experience using PT, Reaper, Studio One, and Samplitude, I don't really think a DAW has a particular "sound", I do believe how you approach it, workflow, plugins, etc.. has something to do with how your project turns out, the only DAW that might actually sound different that comes to mind is Mixbuss, it's designed to emulate an analog system though, where the others aren't

 

LOLZ!! Still I like your avatar more.....

 

What I look for in a DAW is something with a deep purple colour, along with aromas of rich dark currants, nectarine skins, gushing blackberry, but with lots of fragrant tobacco, rich soil, and white flowers, smashed minerals and metal. The best DAWS are medium-bodied with saucy but racy acidity, stabilising the DAW nicely with the robust tannins. The best daws require red currants and ripe cherries, laden with mocha, loamy soil, charred herbs, pencil shavings, and roasted hazelnut. Daws need dense like characters that make them perfect for cellaring, however they are usable straight away once you expose the DAW to the earth’s atmosphere.

 

Well,the question was if DAWS colorize the sound all by themselves (by rendering and by playback processes).
Of course, you can colorize the sound by plugins that do it overtly (and even boast with it), or by those that are trying to hide it (from public and from the non-tinkering user).