It used to mean a deep, spiritual relationship with Nature, sometimes via the "precious balm" of the poppy. As much was true for my good friend Novalis and others who spent their days talking to the trees, Grecian urns or leaves of grass (my ass)!
It means to me a desire to connect, to be connected to the infinite not necessarily through the individual person but through the individual moment. Not knights in shining armor or long-stemmed roses or the dreaded diamond.
Like Novalis, I am at the mercy of a passionate heart and an over-educated brain. The groundswell of my emotion is like the deep Pacific whipped by rattling thoughts as if by a hurricane. For me, the romantic connection is to be able to stare into the depth of that ocean with out the din of the storm. It's peaceful contemplation.
Whether I am looking at the city skyline or the cracked blue turquoise of the subtropical ocean, or into the eyes of a man, I am stirred by millions of bucking, meandering thoughts. In every way possible I try to become one with the moment; to remove the barrier that my consciousness has constructed in order to make sense of things. Being part of the moment is to float above sense, outside the jurisdiction of meaning with no regard for the rational, logical or sensible.
I guess I can see why the "enlightened" ones (by which I do NOT mean the Bodhisattva) saw Romanticism as the enemy of Reason. It pierces through Reason, it shatters its illusion of Truth, it sees that there is only the truth of the moment and that moment is singular.
The cruelty of the Romantic connection is that it is as fleeting as the moment itself.