Engels' devotion to Marx seems extreme, even for a close friend. Engels not only financially supported Marx throughout much of his life, but also went to a lot of trouble to edit and publish Marx's works after his death with no recompense. There is some evidence from Marx's letters that Engels even went so far as to claim an illegitimate child of Marx's was his own child to spare Marx any negative public perception and financial burden.
While it is certainly possible that Engels simply felt extreme friendship and closeness with Marx and it went no further than that, there is no evidence that Engels did this sort of thing -- certainly not to this extreme -- with his other friends and comrades in the communist movement.
Is there any evidence in the historical record, or in Engels and Marx's letters or the like, that Marx and Engels had a romantic or sexual relationship at any time, or at least that Engels had unrequited romantic feelings for Marx?
The answer is no. There is no such evidence, and there is nothing that even comes close to suggesting anything romantic between the two. The way Engels and Marx wrote to each other, was well within the norm for how men wrote to each other at that time. This idea that Marx and Engels should've had some homoerotic affair has no foundation in reality. Furthermore, compare Karl's letters to Jenny with his letters to Engels; Marx is far, far more romantic and loving in those letters, writing poems, describing her beauty, his longing, and so-on, even in periods where they were separated in less than a month. To Engels, you find maybe a "Dearest Friend" or "Life-friend," normal forms of address among 19th century German friends. Maybe signed a "yours truly" or "your friend," but nothing nearly as loving as we see with Jenny. The same goes the other way around; Engels wrote to Marx well within what was acceptable for the 19th century gentleman of Engels' standing. In addition, Engels was a known to have little respect for marriage, and had plenty of extra-marital affairs with other women; something that was at odds with Marx's far more romantic ideas about love, and at times meant that Jenny Marx wouldn't talk to him, if he brought his mistress to the Marx's home.
On the story of the illegitimate child, I assume you're referring to the child of Lenchen, aka Helene Demuth, the Marx family housekeeper, and Friedrich Engels. There is nothing that proves that Karl Marx should be father of Friedrich Demuth, the child, and Engels himself claimed fathership. We also know that Lenchen and Engels were each others confidants, that Engels wasn't the type to stay within the confines of his marriage, and that the two, later in life, formed a professional couple, along with Lenchen moving to the Engels estate, to take on the position of housekeeper. I'm repeating myself here, but there is nothing that, in any way, supports the claim that Marx should be the father of Friedrich Demuth.
The final point I want to raise, is that the idea that Engels' (and Helene Demuth's) work with publishing Capital should be an expression of some love, makes no sense. Engels and Marx were lifelong companions and comrades, they worked on many of their works together. It was only obvious, that Engels would organise and publish the two other volumes of Capital. To claim that it was an expression of love, would you also say that the young Karl Kautsky, who met Marx in London, had some feeling of romantic or sexual attraction towards the older Marx, since Kautsky was responsible for publishing Theories of Surplus Value?
Sources:
III. Abteilung (Briefwechsel) of the MEGA
Liedman, Sven-Eric. A World to Win. 2019. Verso
Green, John. Engels: A Revolutionary Life. 2008. Artery Publications