Google Tracks You
It is simplest to accept that Google tracks as much behavior as possible across all their products and on all platforms. Some of this data is easily tied to particular Google accounts and/or devices. Some of the vast troves of data they aggregate and have little interest in focusing on the individual users.
Youtube URLS
There is no reason to believe that parameters in Youtube video links shared from an app include info that is uniquely traced to the sharer. They do include parameters that can change the way a video is played and also may store info about the platform the video was shared from or other things that would be of interest to data mining.
I cannot duplicate your experience on my mobile device or desktop. I have tested in Youtube Android App (17.39.34), Youtube.com in Android Chrome, and Youtube.com in Windows Chrome.
In all cases, regardless the Youtube URL I start from, all my shared links are shortened to https://youtu.be/[video_id], (https://youtu.be/31NRMIi4rmsas using your example).
I have seen only two versions of si=
values shared across URLs from different users. They are EMSIkaIECMiOmarE6JChQQ
and ELPmzJkDCLju2KnD5oyZMQ.
Your sample URL "https://www.youtube.com/watch?v=31NRMIi4rms&feature=share&si=ELPmzJkDCLju2KnD5oyZMQ" contained the second.
Broken URL Shortening?
I suspect the presence of the long URL with the SI
parameter is a symptom of a URL that failed to shorten. It is peculiar that you would be generating such a long URL from a mobile device. https://www.youtube.com/watch?v=31NRMIi4rms&feature=share&si=ELPmzJkDCLju2KnD5oyZMQ
is 83 characters, whereas the shortened version of that link https://youtu.be/31NRMIi4rms
is only 28 characters, much more suitable for social media, Twitter, etc.
Google Search for repeated values
You can see some similar URLs in Google searches however 500 - 1,000 results are almost a rounding error. Search 2 includes the string that you are generating.
search ?
feature=share&si='x'
- x="EMSIkaIECMiOmarE6JChQQ"
Search only Videos Search all Google
- x="ELPmzJkDCLju2KnD5oyZMQ"
Search only Videos Search all Google