That he later on got engaged to Princess Olga of Greece and Denmark is a well-known fact -- although the exact details of their courtship is murky, as is the reason for the breakup. This happened in the 1920's, long before his eventual marriage to Princess Ingrid of Sweden.
Being linked to numerous women has been nothing unusual for royals -- especially heirs to thrones: Frederik would certainly have qualified as a prize catch on the marriage market. Indeed, the desire for a crown for his granddaughter was the precise reason why King Gustaf V of Sweden discouraged any liaison between Ingrid and the Duke of Kent (fourth son of King George V of Great Britain).
I know that in 1913, there was a get-together of the royal families of Russia and Romania: the purpose was to introduce the Grand Duchess Olga and Prince Carol. It was not unlike the 1853 meeting in Bad Ischl of Emperor Franz Joseph of Austria and his first cousin, Duchess Helene in Bavaria: everybody knew back then that the purpose was to get the two engaged. It's just that the parents hated to coerce a marriage against the couple's will: as such, they were to be given vetoing power. What they hadn't counted on was the emperor using the said power to get engaged to Helene's younger sister Elisabeth instead ...
Flash forward to 1897, and one had that Prince Carl of Sweden and Princess Ingeborg of Denmark got engaged without their own knowledge: it was announced in the newspapers. In fact, the two didn't even meet until their wedding day!
But things had definitely changed by 1913, and Carol and Olga were certainly not going to get engaged by the fiat of their parents. Despite the risk of the latter carrying the hemophilic gene, it would have been an otherwise good match, dynastically and religiously. What hung in the balance, then, was *love*: if the young couple was attracted, then surely there would have been a marriage -- with the risk of hemophilia being set aside.
However, there was no interest: Olga certainly would not have been crushed had her sister Tatiana set out her cap for Carol, instead -- as Sissi had somehow managed to snag the Austrian emperor away from Nene. As it was, Tatiana wasn't any more interested than her older sister.
She did, however, reciprocate the feelings of Crown Prince Alexander Karageorgevich of Serbia: evidently Tatiana was his early love interest -- even giving him a ring.
What all this means, then, is that but for the lack of romantic attraction between Carol and Olga, but for the tragic massacre of the Romanovs by the Bolsheviks, Olga might have become the queen of Romania, Tatiana might have become the queen of Yugoslavia, and Maria might have become the queen of Denmark!
Or WAS it Maria? Did Frederik develop an attraction to one of her older sisters instead?