It is very unlikely. If we assume that Kyle isn't a pathological liar, there is no reason to believe that he is actually John Connor. He speaks of John exclusively in the third person (i.e., "he", "him", "John", etc).
What you describe as him inexplicably being very focused on the photo of Sarah is actually pretty easy to explain in the context of the movie. Kyle is from a post-apocalyptic future where a war of annihilation is being waged against the machines. He's a soldier, and probably doesn't see a lot of women in his daily life. In the few scenes in which we get a look at life in Kyle's time, we see very few women, and the ones we do see are kind of gross and covered in filth. I can't imagine that they bathe on a regular basis, and they probably own a single set of clothes, which are also rarely, if ever, washed. They are traumatized by the constant threat of death at the hands of nearly invincible robot assassins, and spend their lives huddled in bunkers and surrounded by garbage, death, and horror.
The picture of Sarah is a glimpse of a woman in the prime of her life, and in a world where the constant horror and suffering in which Kyle has spent his entire life is totally unknown. She is fairly attractive, she is clean, she is healthy, she is vibrant, she is smiling, and the sun is shining. She seems to be enjoying a moment of peace and tranquility, at ease and free from any apparent sense of danger, which is something that Kyle has never experienced before.
Everyone Kyle knows is always on their guard, all the time. It seems likely that he has never met a person who wasn't suffering from some degree of Post Traumatic Stress Disorder. He has spent his whole life surrounded by broken people who don't know what it is like to be comfortable and safe. People in Kyle's world don't know what it means to be happy. The closest they get to happiness is not being dead yet.
The photo isn't just a picture of a pretty girl. It is a tiny glimpse of a better time and place, a world where people aren't being wiped out by Terminators. A world where people live normal, happy lives.
John Connor saved Kyle's life, and Kyle has worshipped him ever since. John's early life was spent with his mother, and very few other people. His mother taught him everything he needed to know in order to become the leader of the Resistance. She is therefore directly responsible for humanity's only hope for survival. I don't mean to sound blasphemous, but it is probably safe to say that John worships Sarah, and Kyle worships John, so Kyle sees Sarah the way many Christians (most notably the Catholics) see the Virgin Mary - the mother of the savior.
Now we need to consider the nature of Kyle's mission: he grew up hearing about the woman who made the savior of humanity what he was. He received the photo of her, and was transfixed by it, and by her. Then his idol, and his best friend, orders him to go back in time to save this woman whom he sees as a saintly figure, and protect her from a threat she can't possibly understand. This is the most important thing anyone could ever be asked to do. He has to ensure that the last hope for humanity is born.
In this light, his fixation on Sarah, and the photo of her, is only natural. He's supposed to save the entire human race from annihilation, so the task couldn't be more important. The woman in question couldn't be more vital to the future of our species' existence. And Kyle fully appreciates all of this, because he understands how catastrophic the consequences will be should he fail to accomplish his mission.
When you're told to save the only person who can give humanity any chance of survival, you don't take it lightly.
And on a side note, I don't think people would be willing to follow a man who they know impregnated his own mother. That sort of thing is more likely to make everyone hate John than accept his leadership. Gross.
I wouldn't normally make reference to the movie Terminator: Salvation, because it is pretty bad, but we actually see John meeting Kyle for the first time in that movie. This leaves little room for doubt - they are definitely, unquestionably, indubitably, two separate people.