A piano generates a sound like any other instrument. In Music Synthesis we call the shape the sound makes the ADSR pattern (Attack, Decay, Sustain, and Release).
The original question is about the release on a piano.
Ignoring the technical aspects of the piano action, let’s focus on the two main ways you can release sound on a piano.
1) You depress the key with your finger, and you pull your finger off the key.
The way you pull your finger off of the key can have a large impact on the sound. Slow, fast Music, it doesn’t really matter. It is about how long you keep you finger engaged within the millimeters of release space you have access to. The goal with this is to shape the ADSR pattern in a way to your liking, just as a brass or woodwind player would do. It can create an abrupt sound stop or a sound that sounds like it is being muffled (from the dampers/hammers/whatever you want to call it). There is an entire world of color to be had in the release of the sound. Do not believe anybody that says it makes no difference.
2) You do the same thing but you depress the sustain pedal and release the sound with the pedal.
Here the release is controlled by the foot pedals, which can create all kinds of changes to the ADSR patterns. Play a note with the pedal fully depressed, then remove your foot as fast as possible to get a sense of the sound. After that, do it again, but this time release the pedal as slow as you can so it is fully released but you still have some sound. You should hear a large difference in the color of the tone as the dampers get closer to the strings.
Between using the release of the individual attacks to using the pedals to control release, one can create a huge variety of colors on the piano. So yes is the answer to the original question in that how you release the piano affects the sound.
Here is an article talking about the Bach Prelude in C that contains two recording. The second contains a great example of how the sound is impacted by the release. It is easy to hear. Recorded on a Yamaha C7 concert grand piano.