There are lots of different arguments you can pass to rank; it looks like you can use rank("dense", ascending=False) to get the results you want, after doing a groupby:
>>> df["rank"] = df.groupby("group_ID")["value"].rank(method="dense", ascending=False)
>>> df
group_ID item_ID value rank
0 0S00A1HZEy AB 10 2
1 0S00A1HZEy AY 4 3
2 0S00A1HZEy AC 35 1
3 0S03jpFRaS AY 90 1
4 0S03jpFRaS A5 3 5
5 0S03jpFRaS A3 10 2
6 0S03jpFRaS A2 8 4
7 0S03jpFRaS A4 9 3
8 0S03jpFRaS A6 2 6
9 0S03jpFRaS AX 0 7
But note that if you’re not using a global ranking scheme, finding out the mean rank across groups isn’t very meaningful– unless there are duplicate values in a group (and so you have duplicate rank values) all you’re doing is measuring how many elements there are in a group.