I think this is a good use of accumarray().
%some sample data, to illustrate approach
Data = [1915 1 1 1000 1010;
1915 1 2 1100 1050;
1915 1 3 2000 2100;
1918 1 1 500 800;
1918 1 2 1000 600;
2006 12 31 2000 2100;
2008 12 15 1800 1920];
%break into variables
Days = Data(:,2:3);
StreamA1B = Data(:,4);
StreamB1 = Data(:,5);
%get unique month/day combinations. Note, not all months will have the same
%number of days! Leap years can also complicate things a bit.
[MonthDays, ~, index] = unique( Days, 'rows');
%calculate median by day
StreamA1B_med = accumarray(index, StreamA1B, [], @median);
SreamB1_med = accumarray(index, StreamB1, [], @median);
