Thankfully I don't have this problem in my current job, but I did quite some years ago.

It was more or less the same, senior person gets priority.  However a young girl started in my office.  She was not my assistant but I was the office manager at the time.  She argued with me one day that she had every right to take holidays when she wanted despite me trying to explain the procedure.

I swear that people are a law unto themselves sometimes.

If this individual is approachable, go and talk to her, explain the situation.  People should be reasonable in these situations but end up becoming selfish making us resent them.

See how you get on first and then go from there.  Perhaps your holiday period could overlap each other for a few days which wouldn't matter so much and that way you could each still get a week off.