Biochar is not called biochar because of adding biological elements. It is called biochar purely to differentiate it from charcoal, i.e it is charcoal for use as a soil improver, not for barbecuing sausages
That doesn't mean it's not a good idea to inoculate biochar with compost tea, mycorrhizae or any other activating material, I'm just being pedantic about the definition. Biochar is just pure charcoal
When we (Carbon Gold) market it here in the UK we blend it with mycorrhizal fungi spores, wormcasts (a rich source of actinomycetes bacteria) and seaweed and we call it GroChar. There are more and more proprietary blends emerging, but biochar is just plain old charcoal that has been made at lowish temperatures and has definitely not been made with old rubber tires, plastic waste or wood that has been treated with preservatives or paint. It has to be clean feedstock.