My 2001 did not come with one but my 2005 did. If it were me I would not worry about replacing it. What concerns me is that yours has completely self destructed. I find this a little unsettling but I suppose it is possible. It could have just been omitted from the factory by mistake (or they ran out).
If you decide to replace it, you could purchase a new one from a 05 model for $3.79.
Part # : 80030089000
I am pretty sure it will fit your 03.
The material that its made out of is very similar to these 3M Paint and rust stripper discs but more flexible. It is also similar to the black frame padding / filter that you mentioned. It is a type of a very course sponge
3M Paint & Rust Stripper | West Marine
You may just want to do a breather mod (search this site) which plugs the frame opening and runs a pipe with (or without) a one way high speed check valve. This eliminates any possibility of oil mist, or anything else from the motor, ending up in the carb and back into the cylinder.
As an extra added bonus, in the event that you flip the bike upside down, a breather mod will eliminate the cylinder filling with oil.
Another bonus is that you may in theory pick up a fraction of a percent of horsepower by reducing crankcase pressure if you use the one way valve on the breather pipe.
Its too bad we don't have the gallery back up yet as there was a member, Bobzilla, that wedged a Chrysler emissions check valve in his stock breather pipe. Chrysler Part number is 04300376 and cost is about $30.
I use the Chrysler check valve on my 01 but have plugged the frame and run a 3/4 " heater hose out to behind the countershaft sprocket with the check valve spliced in about half way of the length.