Literally anything can be romantic, but watching the sunset on the beach is a prime example of a classic romantic activity, but that does not mean you can't watch a sunset on the beach platonically. What makes a situation romantic is if there is romance. Getting dressed up and going out for dinner for two at a nice restaurant can be romantic or just dinner with a friend at a nice place. Going thrift store shopping can be romantic, or just a fun time with a friend. I would look up some "romantic date ideas" to get a rough idea of romantic themes to inform you better.
Some people expect their partner to make their birthday all about them, that can lessen with age. Not devoting your entire day to your partner on their birthday may be the problem.
If your partner feels there is potential for some sort of romantic involvement with this friend, perceived or real, then you need to talk with them and understand if they know or believe something you don't or if they feel insecure and if there is a lack of trust.
If you didn't tell them what you planned to do, that may be the problem. If you had to ask permission, that isn't healthy in a relationship. Asking them how they feel about you going and making your decision to go, counting their opinion, is wise but understand that you should be able to go do things with your friends and you shouldn't have to ask permission.
If they wanted to go because that sounded like fun and you didn't try to invite them on their birthday, that may be the problem. Just inviting them to come with isn't fair to your friend that wanted to hang out with you and not feel like a 3rd wheel and divide your attention. You can always make plans with your partner on another day and respect the time you have with them and friends.
There are a lot of aspects of this situation that can be why your partner is upset, and it may not be the activity itself. You have to sit down with them and talk to them so you understand their position, they understand your position, and the two of you can come to an understanding and know what may need work in the relationship.