In my story I am writing in third person, my character is kidnaped and finds out his real name towards the end. Do I start referring to him with his real name or the name I used previously?
In my mind, the answer to the question depends upon several factors.
Perhaps the most important in my way of thinking is the identity of the narrator(s). The classic questions about each narrator are: who are they, when are they telling this story, are they reliable, and what vested interests do they have in the outcomes in the story. The "father" as narrator will tell a much different story from the "son". A story told in near real time to the events of the story is much different from the decades-later recounting. A narrator who is trying to justify their actions will emphasize different aspects of the story from a narrator who is "seeking justice" for past bad deeds. Each of these choices can substantially color the common facts. Your question really does not give any clue about what your choices about narration are, and, thus, the number of possible answers spirals out of control.
A second factor is the motivations of the characters. To use the information from your question, does the "son" feel better off as George or as Calvin? How do other characters feel about this? Are there financial, societal, or physical consequences to recognizing the change? Does anyone other than the "father" and "son" know this information? Is it possible to keep this information hidden? Are blackmail and/or shunning in the picture?
We could also toss in writing style. I deliberately referred to the two characters as "father" and "son" rather than commit to a specific name. That is a bit awkward but you might invent a nickname that is independent from whatever the actual name turns out to be.
All of these questions are largely technical, baring on the details of how the story is told. In most cases, the story remains the same. But there is also the question of why you are telling this particular story. How important is the naming of the "son" character to that story?
Perhaps not the simple answer that you were seeking. But the act of writing is never as simple as we think that it should be.
It depends on your character they don't have to assume a name they don't identify towards. I knew someone who changed their name we all assumed they where cohersed into it by their parental figure to gain their love even if it was true the person did not like us calling them by their old name & nick name even now as an adult a slip of the tongue shuts them down.
So this person wants to be the new name for whatever reasons they deem fit it would in essence work that way with your character do they want to be Calvin? Do they want to piss off their kidnappers and use their given name? Did they always hate their forced name? Or what about a third option if they don't want to be the forced name nor the birthed unknown name why not they give themselves a new name and in so doing a new identity they wish to forge in the spirit of the new name?
The media might call him by Calvin for their viewers that doesn't mean he has to let them do so in actual interviews (considering any arise at all). He'd just politely (or not depends on personality) correct them they may to his face call him the forced name for his comfort but print it as Calvin for their readers.
Names are huge deals to the soul it's different from being a kid who tries on the name brings life to who they are in that moment for that role. Your character's feelings need to be your guide here what is their personality like? A name change is a big deal to that person and those around them some people will abandon others for name changes. It changes how people perceive you if you were called Lettice that gives a different impression to another's mind then you being called Ty.
Your character also will need to know who their birth parents are and decide do they want to be theirs? In that changing the name or allowing himself to be called his birth name aligns him with them he is their property, their child, their blood is acknowledged. Did he like his forced name? If he did he'll insist his parents not just temporarily call him this but maintain it making apart of himself always aligned to his kidnapper especially if that kidnapper was someone he admired or liked well growing up in essence keeping the memory bond in this case it maybe love the sinner hate the sin. He may still love the one who raised him this taints his name on one hand it is a bind towards the kidnapping father he loved prior but on the other hand its now a reminder his formative years were at least in parts lies upon lies a pain will always be coupled with the better memories of the lying father.