You really do care about your people and you don't want to come across as thoughtless just because you're in meetings all the time and forget to say anything when a member of your team has a 5-year anniversary or it's someone's birthday.
Just look at the 'Milestones' tab every once in a while. At the top you'll see three important areas: tabs for Birthdays, Anniversaries, and a scope dropdown. The scope is as follows:
Team: My direct reports, my peers, and my boss
Extended Team: Everyone below me in the organization
Department: Everyone in my department
Company: Everyone in the company