In Star Trek Beyond, how could young Spock be holding a photo of an older and deceased Ambassador Spock; are they not the same person?
Star Trek (2009), Star Trek into Darkness (2013), and Star Trek Beyond (2016) occur in an alternate timeline / reality - so there are two different Spocks. "Old Spock" is from the 'original' timeline and has journeyed via artificial black hole to an alternate universe.
The alternate reality was a new reality created when a temporal incursion caused by time travel of the Narada, a Romulan mining vessel from the year 2387, disrupted the time continuum. Accidentally traveling back to the year 2233, Nero, the Narada's captain, attacked the USS Kelvin. The attack resulted in the deaths of several crew members of the Kelvin, including George Kirk and Richard Robau, and the destruction of the Kelvin itself. Spock arrived to the alternate reality in 2258 and was captured by Nero, who used red matter to destroy Vulcan. However, Nero's attacks united the crew of the USS Enterprise, who foiled his attempt to destroy Earth. (Star Trek)
If you are familiar with the Mirror Universe from TOS and DS9 - it's the same kind of concept (although, a different alternate universe)