Why not? The Uncirculated sets I have that range from around 1959 to now, show this kind of die scratches on all the coins in the set. I can't say that I remember specifically seeing an Ike dollar, but they are minted in the same way. Why wouldn't you think this could show up on an Ike dollar? I'm not being sarcastic, I'm really interested in hearing your opinion!
I am interested to know also, if Uncirculated sets that do exhibit die scratches, if a coin with these scratches would be desirable like an error coin having a double date, or Mint Mark?
Die polishing usually takes a circular pattern. Even basining usually only affects the fields, which are the high points on the die. If they were on the die, then every coin from that die would have identical (but in varying states due to wear) scratches. They would also be in relief. Being on every coin from that die, many more would be available for study than the few we see. I find interesting the idea that they are from the roll of stock out of which the planchets are punched. I have never heard of this theory before and believe it deserves more study. I have seen several that were simply stained like this. How or with what, I do not know.
I'll pull out some of the Mint uncirculated sets that I have. I'll i'll see if I can take photos that will pick it up using my iPhone 6. Right now I'm having a problem with my Internet connection on my laptop, so I can't use my Nikon digital to take the pictures which would be optimal! Possibly, if I use good lighting some of the scratches will hopefully, be evident. Actually, the circular aspect of the scratches that you mention, may be what I have seen on them. I'm not sure, as it has been a while since I have looked at them. I'm curious though, are uncirculated sets with the circular type of scratches on them desireable, or is it basically a waste of a good uncirculated set? Thanks
Basining and polishing of dies is to basically extend their useful life. If there are clash marks or other damage, the mint employee can smooth them off somewhat. I have never hear of a premium for polish marks.