Is there a system of automated sharing of current (experienced) weather between aircraft in flight?
Aircraft to aircraft, there is not.
Ground or satellite to aircraft exists in the form of FIS-B, and other commercial providers.

However, aircraft to ground in an automated fashion exists.
It's called Aircraft Meteorological Data Relay (AMDAR).

[It's] a program initiated by the World Meteorological Organization. AMDAR is used to collect meteorological data worldwide by using commercial aircraft.
Data is collected by the aircraft navigation systems and the onboard standard temperature and static pressure probes. The data is then preprocessed before linking them down to the ground either via VHF communication (ACARS) or via the [Aircraft to Satellite Data Relay].—Wikipedia

So, if it becomes widespread, FIS-B, et al., would benefit from AMDAR if they share the data. Thus realizing the aircraft to aircraft weather relay, in a sense.