No, I'm afraid our external tile provider does not include that information in the downloaded tiles. Tiles are updated at different times in different regions of the world, so we cannot assume either that all downloaded tiles were recorded on the same date.
There are geo imagery providers that serve data with precise date information, and offer a range of possible dates, including recent tiles. You would have to contact them directly, obtain a key and then use a GeoServer -> "..." specification to use those tiles from Wolfram Language.
