The actual claim is being dramatically overstated.
NASA Administrator Bill Nelson told Politico that Chinese military aggression in the South China Sea indicated what might happen on the moon.
www.businessinsider.com
A NASA official said that based on how China claims the entire South China Sea and aggressively invades any area with resources, if they establish a permanent base on the moon before the US does they might attempt to claim some of the more resource-rich areas for themselves. China themselves have not said they're claiming the moon as far as I know, and are actually just aiming for a manned mission in 2030.
That said I don't think anybody would be stunned and shocked if China did try to claim territory once they had a working base on it, but heck, if the US, Russia, or the Amish built a lunar mining colony they'd say the mine was theirs too.