Does vinyl have a sound?

Other than great resolution, timbre, and soundstage, can one recognize a sound that vinyl has?

Sometimes I think I hear a "plastic" sound, especially on percussion. Does this make sense to anyone? If so, does this go away with proper choice of table, arm, and cartridge?
I think it's not a great question. There are too many variables in the analog chain that impart their own colorations for one to say per se that "vinyl" has a sound of its own. On the other hand, tape definitely has a sound of its own, which makes it hard to know whether the ability to distinguish one from the other is due to the "sound" of vinyl or the sound of tape.

A poor turntable that blurs rhythm and pitch, a tonearm that is poorly aligned and adjusted, a badly tracking cartridge, a poor tonearm/cartridge match that results in bass distortion, bad or damaged LPs with ticks, pops, crackles: are these (some of) the sounds of vinyl? Since they all can be eliminated or significantly ameliorated, I would say not.
A lot of information and misinformation in the answers. Let's see if we can sort it out.

First, the misinformation. Yes, all formats have some artifact but artifact of LP is not 'plastic' by any means. It has greater dynamic range of the best digital and tape, although it is rarely expressed (the limit is on the playback side, not the record side BTW). The RIAA curve is not a cause of phase shift. In fact the media has the least phase shift as it has the greatest bandwidth, often extending out past 50KHz.

If your setup is poorly done, you will encounter all sorts of artifacts. A good cartridge is key but it need not be expensive if the arm can make it track correctly. What this means is that the arm is far more important than the cartridge. The platter can affect soundstage; if it is varying its pitch even slightly the arm will jog back and forth over the stylus which will impart a shimmer to the soundstage that tape and CD lack.

The choice of phono preamp can affect how much you hear in the way of ticks and pops; this has nothing to do with bandwidth and everything to do with the stability of the phono section!

If you have a poor setup the cheapest CD player may sound better; if you have good gear with good setup you will find the performance hard to beat with any other media. The truth of this last statement is why you see such variable responses.
I have read reviews where they describe 'cardboardy' sound on poorer set ups. I think you may mean this when you describe plasticky. In this instances the sound is very two dimensional- no depth perception at all. The record noise is major contributor in addition to not having proper coupling with the platter. I personally have found this when there is not enough center clamp force or the right clamp material. just my 2 cents.
I think its reasonable to say that there is a lot that has to go into vinyl playback both in terms of setup and cost in order for it not to have a recognizable sound. Also, you have to have very good source material in very good condition.

This is the exception found only in a small % of all vinyl setups out there, these mostly owned by dedicated vinyl audiophiles with the significant knowledge needed to get there (or the $$$s needed to pay someone else with the needed knowledge to help get them there).

The rule, is most vinyl on most tables out there will be less than optimal setups and easily recognizable as vinyl as a result ie have a distinct sound.

Back to my earlier statement that the sound of good vinyl and good CDs tuned to the same reference standard can be hard to distinguish. But in practice, this is the exception not the rule.

I think these days one must be willing to put $2000-$3000 dollars minimum (plus significant time and expertise to set up and maintain properly, including record cleaning) into a vinyl rig to have a chance of accomplishing this.

The cost for similar digital is much less. It can be done very well for < $1000. Plus setup and maintenance with CDs is minimal once you get the right gear. Music servers are more complex, but not a big deal if one is already computer literate, a good skill to have in general these days (an added bonus).

I am a vinyl fan, but I will not sugar coat what it takes these days to get the results that many expect based on the hype. Its much easier and cheaper to get there and stay there with digital these days.
Nil, In connection with the recent passing of Dave Brubeck, I took out some of my old Columbia LP recordings of the DBQ. Gone With the Wind is a 6-eye that I have owned since new (i.e., >40 years), played it on any and every rig I have ever owned, good and bad. The LP is full of ticks and pops, but the dynamics and sense of depth are completely intact, still thrilling in fact, if you can listen past the surface noise. Paul Desmond's solos are still to die for. So, this one example would lead me to say that the sense of depth and 3D-ness that are so characteristic of the best in vinyl are not an inverse function of surface noise at all. On the other side of the same coin, some of the recent re-issues of classic LPs are disappointing in these same areas compared to the originals, yet they are dead quiet.