I'm currently waiting for a final interview with an employer. But today is his birthday that I knew about from his Skype profile. I wonder if I should send him my birthday wishes or is going to look like I'm flattering him?
No, you shouldn't.
Given that you just know his birthday by mere accident, in this case from his Skype account, there is not really anything to gain from it. If you do, it could be perceived as kissing up. However, it is very unlikely you will get any "bonus points" for sending him best wishes.
As many people don't check the Skype birthdays or turn the alerts off it would not be considered weird if you simply didn't know his birthday and hence don't send him best wishes.
I would consider it weird and creepy. I wouldn't worry, I can handle weird and creepy, but I can do without it. In some respects, I might be worried that I'm inviting a stalker to my company and god knows where that might lead.
In both cases, that final interview might not happen, and if it happens, it might very well fail.
I seriously suggest that you wait about a year, and if there is a birthday card going round that everyone signs then you join in and sign it as well. Right now it doesn't look like you are flattering him, it looks like an invasion of privacy.
PS. I don't tend to use my true birthday when websites ask for it. So if you sent me a birthday card on the day that is supposedly my birthday according to Skype, it's unlikely to be my birthday.