I suspect most people think port and starboard are just synonyms for left and right. I did for a very long time. And you also have to consider non-native speakers who might not know the nautical terms at all.
The trains where I live say "on my right" and "on my left". It sounds a little weird hearing an automated voice speaking in the first person, but it's concise and gets the message across to people who won't know nautical terms.
Sure that would be technically more correct, more tasteful, and less confusing. But some people may not know the words "port" and "starboard." Frequent train users would become accustomed to it soon enough but you'll always have some people from out of town whose first language is not English, and their limited vocabulary is more likely to include "left" and "right." On a utilitarian basis we must weigh their needs in the equation when considering this problem. Then again there are also some people who can't tell left from right either. We could paint one side of the train green and the other red, but then we might have a problem with the colourblind train users' lobby. What's really needed is 15 million dollars and a six-month user experience design study to decide on the best approach.
Where I’m from, nothing announces the exits. There’s just a pair of doors opposite each other and you go out whatever one opens. I’ve never even thought about it before. Is there a reason it needs to be clarified on your trains?
In my city people often stand with bikes out strollers in the door area. Out sometimes it's just really packed. When the opening side is announced, those that stand in the way can already make way (if they are considerate to their fellows) or know that they can relax and don't need to move.
It’s super important in Japan where jam packed trains are common during rush hours. They typically announce in Japanese and English, and may even have a visual displayed.