Worth noting is that Princess Alexandra received the wedding gift not from her parents, but from King Frederik VII who hoped to impress the British Royal family. The King saw the marriage as the best chance for the Danes to get British support against the Prussians in the struggle for Schleswig-Holstein. I've found Danish mentions from the time that calls the necklace wasteful and even ridiculously extravagant.
At the time of Princess Dagmars marriage to Tsareivic Alexander her father had become king,but her parents chose to not attend the wedding in St Petersburg because they were embarrassed about looking to poor compared to the Romanovs so I doubt that their gift was as extravagant as the Dagmar necklace. The new King and Queen had a limited private fortune and with the exception of the Crown jewels Queen Louise had few significant pieces of jewellery herself.
I have found that The Princess received an elaborately decorated set of books and a painting of flowers by the artist Anthonore Christensen, but I don't know by whom.