This depends on the vehicle.
EVs that are based on the electronic platform of current ICE vehicles are using conventional OBD2 standard. This is valid for cars like VW e-Golf. Those vehicles can have brand-specific messages that can be decoded with a proprietary diagnostics system (including those ones that are specific to the electric powertrain). Those parameters are monitored, stored, and processed on ECUs that are responsible for a specific part of the system (such as for example DC/AC or DC/DC converter, electric powertrain controller, battery monitor control unit).
Most of the vehicles use both the OBD2 connector and ISO 15765 protocol for communication. The drawback of this is that standardized diagnostic code readers have implemented codes that will not be utilized in the EV (such as the P01XX group for fueling). On the other hand, EVs are dependent on other parameters such as SoC (State of Charge), cell temperature, cell heating, or cooling status, which were not an aim during the initial onboard diagnostics system design.
There are vehicles like Tesla which use solutions that are specific to their own brand. As far as I know, some Tesla vehicles (like Model 3) use adapters that can convert to OBD2 connector.
This is an interesting topic, as according to EU legislation, M1 category passenger vehicles needs to be equipped with EOBD standard. EOBD standard is vastly similar to OBD2. Petrol cars registered after 01.01.2001 and Diesel powered vehicles registered after 01.01.2004 need to comply to get EU homologation. I believe that during this legislation electric vehicles were not included, therefore this may not apply (I need to investigate it more. I am not fully sure about this one).
Also worth noting is the fact that EV exchanges information about some of the mentioned above parameters with charging station. This information are crucial to safe and effective charging. A good source of information regarding this can be found on Open Charge Alliance. For more information please visit: Open Charge Alliance