I have one also, it has a liner for cold food and yes you could do ice but its also a hot food keeper. Yours does have bakelite handles though making it an earlier one (mine too)
I have the identical Ice Bucket with the lid, but the thermos part broke, (in between the walls), and now it rattles! I remember when my dad dropped it and my mom had a fit! I inherited it and found it will hold cold salads and stays cold a long time while on table because I refrigerated it overnight! So, I still enjoy it!