Is the prohibition on homosexuality only on the act of sodomy or is it for all romantic relationships of this kind?
Committing the act of sodomy is a sin and a crime that is comparable to adultery.
However, just like for Zina, things that lead to this crime are also forbidden. Because Allah says for Zina:
And do not go anywhere near adultery: it is an outrage, and an evil path. (17:32)
The same concept applies (doubly so perhaps) for sodomy.
Furthermore, as Crimson quoted, the Prophet Muhammad (SAW) said:
A man should not see the private parts of another man, and a woman should not see the private parts of another woman, and a man should not lie with another man under one covering, and a woman should not lie with another woman under one covering. (Sahih Muslim)
This clearly forbids any romantic relationships between the same gender. Not even sleeping under the same covering even if they do not commit the crime itself.
So, in fact, there is no doubt that the action is a sin.
However, at the same time, a person is not automatically sinful just because they have desires to commit sins. Everyone has desires to commit sins. And if we do not act on it and choose to ignore the desires, there is no problem:
Allah has forgiven my followers the evil thoughts that occur to their minds, as long as such thoughts are not put into action or uttered. (Sahih Bukhari)
So, even if you have desires for something evil, that does not mean you yourself are evil.
There are more kinds of love than romantic. If you are incapable of feeling romantic love normally, devote yourself to loving Allah. There are many people who are incapable of being married due to whatever reason even if they want to. That does not mean those people are allowed to fall into adultery.
Rather they must control their desires, and be patient for what Allah plans for them even if that means never marrying.
Actually the Quran makes no mention of romantic relationships involving members of the same gender. Allah is The most merciful and aware of our struggles, however we are required to abstain from the haram, and He loves those who are repentant from sin. What is mentioned by Allah swt is that the people of Lut approached other men with desire instead of women, one of their many sins. The issue here is that people broaden the word "approached" to mean any action with desire when Allah swt uses this word to also tell believing men not to approach their wives when they are menstruating. In a hadith prophet Mohamed sheds some light on this topic, by mentioning that he would fondle his wife while she was menstruating, after advising her to wear a waist wrap. Approach means intercourse, in the context of the people of Lut, their sin being described is sodomy (intercourse from the rear area). I think Muslims stretch this prohibition too far when they label even holding hands as sinful between a gay couple. It's important to note the restrictions however, such as maintaining modesty and not falling into sexual encounters, but a gay male does not avoid lwat(sodomy) the same way a straight man avoids zina with a woman, something clearly mentioned by Allah swt. Example a gay male is not forbidden to shake hands with another man even if he finds him attractive. One Muslim once advised a gay male not to go to the mosque if he finds there to be fitna, missing out on the good deeds and obligations.
My conclusion with this, is that, a gay Muslim couple should only be together in company so that they can avoid haram temptation. Kissing, hugging, and holding hands to my knowledge is not prohibited since these can be done non sexually, out in public with modesty, and are just actions of affection and love. A gay male should lower his gaze from provocative sights and the awrah's of people, same as any Muslim. I understand how Muslims may see even this as stepping too far, but gay Muslims can remain celibate while still being with a same gender partner they love and can grow old with. The love is not haram, the sexual encounters that step the bounds of modesty are.