They could have meant 'tone' or 'musicality'. So, as you say in your comment, you can be 'technically perfect', hitting all the right notes at the right time, but perhaps not 'feeling' the music. Also like you say in your question, listening to great recordings is a good way of hearing good musicality and tone! However whether you like someone's specific tone or not can be subjective.
Having said that, there are things you can do to help you develop your own special tone and you musicality. Do you ever record yourself and listen back? If not, then try it, it doesn't have to be a great quality recording, even just using your phone/computer's basic record function to hear how you sound can be a great help! It can be embarrassing to listen to yourself... (like listening to your own voice) but try it! Then see if you think you sound 'wooden' or 'lacking feeling' in certain phrases, and try and add more feeling. Find a recording of a piece you love and try and emulate that sound. Listen to lots of versions of the same piece to find one you love, and try and copy that style. Eventually you will be able to define your own style.
Also try playing with other people if you don't already, either accompanying someone (this will depend on what instrument you play) or jamming together can help you modify your style for different situations, so that you develop that 'tone' and 'musicality' control over your playing. This way, you will automatically learn how to blend in with other sounds, or to stand out (for example if you have a particular phrase that you should accentuate at a specific point in the music, like a solo).
This is an important part of learning how to control your instrumental tone, but when you get it right it will be really satisfying!