The answer is:
Yes!:
As you can see here, with this pattern you can easily fill the 16x16 grid. In every colorthe top left corner, youI have filled in the mini sudoku in pink. In the other 3 corners, the 4 pink squares are able toplaces in such a way, that no pink square aligns with a pink square in another corner. That way we can fill in the pink squares with exactly the same numbers as the mini sudoku in the top left corner. Basically still mini sudoku's, but taken apart.
The other 3 colors work exactly the same, and could be filled in with the same numbers. For example, you can put in the same way as 1,2,3,4, the 5,6,7,8 in the yellow squares, the 9,10,11,12 in the green squares and the 13,14,15,16 in the purple squares.
@John Bollinger shows in a really nice way how many different solutions there are, just with this setup: We are effectively dividing the 16x16 grid into 16 independent 4x4 puzzles, split evenly among 4 color-coded categories. With 288 distinct 4x4 sudoku (including label permutations) that makes for 288^16 total ways to fill out such a grid. Even if you divide by the 16! permutations of the labels, that still gives you about 10^26 solutions of this form.
Bonus: you now have 4 mini sudoku's in 1 big sudoku.