I forgot to point out that Schala said that when they find each other, they may not know that she is Schala and Serge is Serge. This could have infinite possibilities as to how this could happen and how they wouldn't recognize each other. Maybe this is a possibility as to why Serge doesn't know who he really is.
I have even considered that Serge could have been the product of something similar to Kid (like Project "Serge", with some connection to Janus), but I haven't figured out a way to explain how that could be possible. I think it is interesting, though, that the picture that is shown at the end does not reveal exactly who is standing next to Schala. I mean, if it Serge himself, why not just show that it is Serge and not give it a second thought? Why would the developers "hide" the person standing next to her?
I have also always looked at the Chrono series as the story of the Zeal family (because most of the significant events of the whole series ocurred in Zeal and had a direct connection to the Royal Family), so it would make sense that Serge would have some underlying connection with Schala.
Perhaps, or yet Lavos' tale. He is, after all, the only thing that is fully constant through all its incarnations.
But anyway, it really is quite impossible that Serge is Janus, and the connections are really only superficial. As someone pointed out, it is beyond doubt known who was to be Janus, and that was Guile. If you've ever played Radical Dreamers, it shows the distinct characters of Magil/Janus, Kid/Schala, and Serge. However, for whatever reasons - time constraints, I believe - they opted to cut that out, and we were left with a Guile being simple a wandering magician. It is not inconceivable, however, to my mind, that if CB is ever made, they will come back and prove that Guile and Janus were one in the same - they just didn't have the chance in CC. There is, after all, nothing that disproves them being so. And we must all remember the elusive shadow on the ground if you take along Guile to Viper Manor when Luccia gives Kid the letter...
Anyway, it is true that Serge was part of a project, but it was of a different sort - he was the trigger to initiate project Kid or, rather, allow it to be. You see, the whole idea of the project was that Belthesar, perhaps seeing Schala still as his princess, and thus owing oaths of fealty to her, attempted by very complex means to effect her rescue: this was project Kid. Schala herself aided in it in the creating of Kid; Serge did so by splitting the dimensions, and through that forging the requisite Chrono Cross, the only means by which she could be saved.
As for Serge being something similar to Kid... well, it is very plain that he is indeed the son of Wazuki, and no fatherless clone, as is Kid.
Really, one must look at how much the possible differences outweigh the similarities, and then take into account what the gamemakers intended. Serge shares almost nothing with Janus. Their moods are different. Their builds are different. Their hair is even different, in all likelihood (well, Janus' is light blue, and Serge's dark blue; it is possible, as they likely both dye it - Janus does so for a certainty, according to the game - that their natural hair colour is both black, but that's speculation.) And, if truth be told, Serge, in build and style of fighting and all, is most similar to Crono. Do they not both share Luminaire and slash-type attacks, and bear an innate colour of white and heavenly sorcery? And as for maintaining that he must be Janus because she would not care for a stranger otherwise.. Schala was ever-compassionate, that much is plain even in Zeal. She was likely only moved to pity by his cries that echoed to the Tesseract, sort of a strange love-at-first-sight.
As for your first comment regarding Schala saying they might not know each other... that's just bringing in reincarnation, and a shlocky thing that the player might be the reincarnation of Serge into the whole thing. Remember the end video where Serge is searching through the dimensions, even across the modern world, our world, for Serge? That's all it is, and really has no connection to Janus whatsoever.