re: I would like to add the year in de date format, but that will show the current year and not the year of birth. Is there a way to show the year of birth?
Sorry, no, the list is designed the show the upcoming anniversaries/birthdays, not the actual anniversary/birthday year.
re: I also would like to add a link to the entry. Is that possible? Or should I add a widget shortcode on a nromal page to do that?
Sorry, no, the upcoming list shortcode can not link to the entry. It is possible with the widget shortcode though. If you do decide to use the widget shortcode, it does have different options and a different display format so it will not be exactly the same as the upcoming list shortcode.
I hope that answers your question thoroughly!