ORAC, there were two separate maritime Task Groups in the Adriatic back in the early 90's: a WEU one (Sharp Fence) and a NATO one (Maritime Guard) which eventually merged into Operation Sharp Guard. (A joint Op, as it were).
It is not a surprise to see the French and EU members have an op that is scoped to their needs only. In time they may choose to merge with the USN+Japan+et al operation, but politically there may be constraints or objectives that don't match at the moment.