Both aquamarines are rectangular, but the diamond setting surrounding the aquamarine is different on Princess Lilian's and Princess Madeleine's rings. Therefore, I think these are two different rings:
I do regret that Princess Madeleine did not coordinate her tiara with other aquamarine jewels that she has at her disposal. For instance, the diamond floral earrings she wore for the Nobel ceremony are lovely by themselves, but their delicate floral design is not a good match, IMHO, with the geometric pattern of the aquamarine kokoshnik. Princess Madeleine owns a pair of aquamarine and diamond earrings whose colour and pattern would have made a much better match with this tiara (just my opinion, of course...):
Princess Madeleine and Chris O'Neill's pre-wedding reception, June 2013: