Is it possible to change the name of the bookmarked page when a user adds my page as a bookmark without changing the title tag?
No, title tags are the way you set your default titles when bookmarking. It's entirely up to the user whether they want to change it from that default.
Sidenote: This question is also more appropriate for StackOverflow as it doesn't directly relate to WordPress (WordPress Answers is solely for Wordpress-related questions). :-)