J, 59 bytes

,&([:(0 #@-.~~.@,)](*@[*[:>./((,-)#:i.3)|.!.0 ])^:_ i.@$)-.

Try it online!

This is one of those where the idea is incredibly simple (and I think fun), but executing it had some mechanical lengthiness which masks the simplicity... eg, shifting the original matrix in all directions with 0 fill is the verbose ((,-)#:i.3) |.!.0.

It's likely this mechanical lengthiness can be golfed further, and I may try tomorrow evening, but I'll post the crux of it now.

Say our input is:

0 0 0 0
1 1 1 1
0 0 0 0
1 1 1 1

We start with a matrix of unique integers the same size:

 0  1  2  3
 4  5  6  7
 8  9 10 11
12 13 14 15

Then for each cell we find the max of all its neighbors, and multiply by the input mask:

 0  0  0  0
 8  9 10 11
 0  0  0  0
13 14 15 15

We iterate this process until the matrix stops changing:

 0  0  0  0
11 11 11 11
 0  0  0  0
15 15 15 15

And then count the number of unique, non-zero elements. That tells us the number of 1-islands.

We apply the same process to "1 minus the input" to get the number of 0-islands.