I have many Muslim friends. Only one of them celebrates his birthday according to the Islamic calendar. His birthday has circled the whole Gregorian calender once in the time I’ve known him.
But age is even more difficult. As soon as the calendar ticks over into a new Islamic year his age increases by one year. I don’t believe the latter is an Islamic thing per se. It’s a practice of his particular sect/ethnic group.
For legal and official purposes he uses his birthday and age according to the Gregorian calendar.