Is Breadth First Search Space Complexity on a Grid different?
It depends what you mean by a "on a grid".

If you mean that you are working with the graph of a complete grid, with number_rows * number_cols vertices (one for each grid point) and an edge between every pair of adjacent vertices:

Yes, that's correct.  The asymptotic time complexity of both is O(number_rows * number_cols), and the asymptotic space complexity is O(number_rows + number_cols) for BFS or O(number_rows * number_cols) for DFS.

If you mean that you are working with a graph that is a subgraph of a complete grid, then:

No, that's incorrect.  The asymptotic time complexity of both is O(number_rows * number_cols), and the asymptotic space complexity of both could be as large as O(number_rows * number_cols), as explained in https://stackoverflow.com/a/50912382/781723.