I must have dreamt about this last night because I woke up this morning with a new thought. OK Lets assume this medallion is post 1938. In a hospital nursery infants are often referred to as Baby Miller etc until the birth certificate is finalized. In this case they listed the mother's name and the sex. Perhaps it was pinned on the crib in the hospital nursery with a thumbtack.
The other possibility is a mortuary tag for a stillborn child that was unnamed.
Perhaps Madeline was a hereditary name in the Miller family? Might check Ancestry.com, or death records in the county, post 38 under Miller. When children were born mostly at home midwives were unable to save some infants. The child could have been buried in the back yard along with this tag. There may not have been a record.
How about this one?
Name:
Mary M Miller
Death Date:
15 Jan 1952
Death Location:
Cook County, IL
File Number:
6001473
Archive Collection Name:
Cook County Genealogy Records (Deaths)
Archive repository location:
Chicago, IL
Archive repository name:
Cook County Clerk