I got the solution. The searching works with the item.value, not with item.child. I prepared the "DropdownMenuItem" like below:

DropdownMenuItem<String>(
    value: "$rawmatCode-$serialNocapture-$rawmatName2",
    child: Text("$rawmatCode ($rawmatName2)"),
      );
}).toList();

And the search match function works like below:

searchMatchFn: (item, searchValue) {
   return item.value.toString().toLowerCase().contains(searchValue);
},