In heraldry, a cognisance (or cognizance) is a particular symbol used by people employed by or otherwise affiliated with a particular house, and also on the property of that house. It is not necessarily the same as a coat of arms, but may share some symbols with it.
A heraldic badge... indicates allegiance to, or the property of, an individual or family. Medieval forms are usually called a livery badge, and also a cognizance. They are para-heraldic, not necessarily using elements from the coat of arms of the person or family they represent, though many do, often taking the crest or supporters. Their use was more flexible than that of arms proper.
Badges worn on clothing were common in the late Middle Ages, particularly in England. They could be made of base metal, cloth or other materials and worn on the clothing of the followers of the person in question... The badge would also be embroidered or appliqued on standards, horse trappings, livery uniforms, and other belongings.
The UK College of Arms, which is the body responsible for granting official heraldry in many Commonwealth realms, further describes the difference between a coat of arms and a badge (cognizance) as follows:
While arms and crest are personal to their bearers the badge may be used by others wishing to show connection or allegiance to the individual or corporation to whom it belongs. Thus it is appropriate for the employees of a company to wear a tie bearing the company's badge, but not the company's arms.
So the learned antiquarians are saying that "Bleeding Heart Yard" used to belong to a noble family who used a bleeding heart in their heraldry. The bleeding heart is itself a common element in Christian iconography.