Does vinyl have a sound?

NO,YES/MAYBE

Also, you have to have very good source material in very good condition.

^^^^this^^^^

I've spent the better part of 45 years collecting, cleaning, curating and upgrading my record collection. Much more time and money than my system itself.

I'm constantly on the lookout for better copies or earlier pressings of what I already have in my collection.  Also, pressings from other countries such as the UK, Germany, Russia ,(classical especially), Japan, etc. 

Most modern pressings pale in comparison to a good early pressing, but the search for quiet copies in NM condition can be frustrating.

Find the right pressing with decent equipment and there is no "sound", just the music.

@mechans , the negative comments on unipivot arms come mainly from me I think. Let me try to explain it carefully so you understand the issue and what to listen for. Cartridges are designed to "read" lateral and vertical information translating it into two channels. Normal pivoted arms allow movement in only two directions, lateral and vertical. There is a resonance frequency associated with both directions depending on effective mass and the compliance of the cartridge. In an effort to simplify construction and limit friction tonearm designers came up with the concept of a single needle pivot. There are several problems with this design. First is durability. All the weight of the tonearm and cartridge rest directly on that single point instead of distributing it. Consequently the bearing wears out faster. VPI owners frequently have to change "needles." Next and more important unipivot arms allow for a third degree of movement and that is torsional which changes azimuth but worse adds an additional resonance point. Because the effective mass in the torsional direction is much lower the resonance point tends to be at a higher frequency, into the audio band. This is going to effect bass and there is where you have to listen. I think if you compare a record to it's high res digital copy you will hear it. High end manufacturer's of unipivot arms have gone out of their way to solve the problem. Graham uses an opposing magnet stabilization system and Basis uses what is essentially a second bearing that the tonearm leans against. Johnathan Carr, designer of Lyra Cartridges recommends specifically against unipivot designs. 
If you are happy with the sound of your system then there is no need to change. If you want to make an improvement then this is a good place to do it. VPI makes a gimbal pivot are that is a drop in replacement.
I am not trying to anger people but I think truth is important also. 
Mike

To answer your question - YES.  I was at a presentation by the owner of Genesis Speakers - Gary Leonard Koh owner brought six records to listen to all made out of different material.  Each had a sound.  Very easy to hear the differences.  He is a fanatic so reach out to him and ask him all the questions you have on the subject.

Happy Listening.