I mean, Anakin refuses to leave Obi-Wan behind on that ship, even though it might jeopardise his mission. Their bantering during that entire rescue is also delightful. In AotC their relationship is more tense, but even there we start on the elevator scene, where I think it's made clear that they both love and know each other well. In RotS we also have the extremely warm goodbye scene when Obi-Wan goes to Utapau. After landing Obi-Wan and Anakin is clearly in a celebratory mood, but I don't think it's strange that most of the people that meet them give more attention to how the chancellor, the head of state, is safe. Anakin also leaves before attention could move to him to steal some super secret alone time with Padmé xD
And honestly, it's not the first time they have pulled something like this off. It's not a post Death Star moment of wonder and victory, it's Friday during the clone wars.
I do think a decent argument could be made for marriage not being allowed for the Jedi (in fact that is the view I tend to take myself), as they are already married to the order is many ways. But marriage is of course not the same as a relationship or even having feelings for someone. And at least the feelings part high canon has explicitly established that it's not a problem.
"Jedi Knights aren't celibate - the thing that is forbidden is attachments - and possessive relationships." –George Lucas, BBC News 2002 interview
Based on this quote I would also guess that casual or non-attached relationships would also be okay.
Different materials interpret Jedi culture and what attachment is or isn't in several different ways. As for how it's meant to be understood in high canon (movies and tv-series) I think Lucas's quotes on the matter might be the clearest:
The Making of Star Wars: Revenge of the Sith, page 213:
"No human can let go," Lucas would say of [the Yoda-Anakin scene]. "It's very hard. Ultimately, we do let go because it's inevitable; you do die and you do lose your loved ones. But while you're alive, you can't be obsessed with holding on. As Yoda says in this one, 'You must learn to let go of everything you're afraid to let go of.' Because holding on is in the same category and the precursor to greed. And that's what a Sith is. A Sith is somebody that is absolutely obsessed with gaining more and more power - but for what? Nothing, except that it becomes an obsession to get more."
"The Jedi are trained to let go. They're trained from birth," he continues, "They're not supposed to form attachments. They can love people- in fact, they should love everybody. They should love their enemies; they should love the Sith. But they can't form attachments. So what all these movies are about is: greed. Greed is a source of pain and suffering for everybody. And the ultimate state of greed is the desire to cheat death."
Star Wars: Attack of the Clones commentary track, George Lucas:
"The fact that everything must change and that things come and go through his life and that he can't hold onto things, which is a basic Jedi philosophy that he isn't willing to accept emotionally and the reason that is because he was raised by his mother rather than the Jedi. If he'd have been taken in his first year and started to study to be a Jedi, he wouldn't have this particular connection as strong as it is and he'd have been trained to love people but not to become attached to them.
"But he has become attached to his mother and he will become attached to Padme and these things are, for a Jedi, who needs to have a clear mind and not be influenced by threats to their attachments, a dangerous situation. And it feeds into fear of losing things, which feeds into greed, wanting to keep things, wanting to keep his possessions and things that he should be letting go of. His fear of losing her turns to anger at losing her, which ultimately turns to revenge in wiping out the village. The scene with the Tusken Raiders is the first scene that ultimately takes him on the road to the dark side. I mean he's been prepping for this, but that's the one where he's sort of doing something that is completely inappropriate."
Overall
I would give Lucas's own quotes much more weight regarding how we are supposed to understand terms and themes in the movies than I would give ancillary materials. But nobody has to agree with that.