Sgr A*, M87*,... Several supermassive black holes (SMBHs) at the center of galaxies have a star (*) in their name. But many others do not. Is there a physical difference between the SMBHs that are denoted by a star and those that are not?
How the name came to be chosen is discussed in this paper1 (§3, pp. 4–5):
Eight years after the discovery, one of us (Brown) invented the name Sgr A∗ to distinguish the compact source from the other components in the galactic center and to emphasize the unique nature of this source.
When I [Brown] began thinking of the radio source as the “exciting source” for the cluster of H II regions seen in the VLA maps, the name Sgr A∗ occurred to me by analogy brought to mind by my Phd dissertation, which is in atomic physics and where the nomenclature for excited state atoms is He∗, or Fe∗ etc.”
1 Goss, W. M.; Brown, Robert L.; Lo, K. Y. (2003-05-06). "The Discovery of Sgr A∗". Astronomische Nachrichten. 324 (1): 497. doi:10.1002/asna.200385047.
Sgr A is a radio source from near the centre of the galaxy. In early radio wavelength maps it wasn't clear if this was a single source or multiple. With higher resolution it is clear that Sgr A is actually multiple objects: A supernova remnant, several clouds of ionised gas, and a compact source of radio waves.
Sgr A* was a named (by its co-discover) Robert Brown. He said, later that he attached a * because he thought this object was "exciting", and excited states of atoms have a * attached:
“ Scratching on a yellow pad one morning I tried a lot of possible names. When I began thinking of the radio source as the “exciting source” for the cluster of H II regions seen in the VLA maps, the name Sgr A* occurred to me by analogy brought to mind by my PhD dissertation, which is in atomic physics and where the nomenclature for excited state atoms is He*, or Fe* etc.” (source)
On this model, some other supermassive black holes are also designated by a *. But there is no systematic naming of black holes like this. Many don't seem to have their own name and are merely "The SMBH at the centre of ..." Other black holes were previously known as quasars, and have a quasar designation, or have multiple designations as radio sources, X-ray sources, UV sources. And so appear under different names in different contexts.
Asterisk generally indicate that the body is located at the center of galaxy. For e.g. M87's black hole or "M87*" is located at the center of Messier 87 (M87) galaxy. The reason for this notation is explained very well here by taking the example of Sagittarius A*:
The name 'Sagittarius A*' refers to the believed location of the supermassive black hole in the center of our galaxy. About 50 years ago, astronomers identified an area within the constellation of Sagittarius that was the strongest region of radio emission thus making it the likely center of the Milky Way. Astronomers dubbed this region 'Sagittarius A' to identify this particular area within the large patch of sky that this constellation covers. As science and technology improved, astronomers were later able to pinpoint a more exact location of the suspected location of the supermassive black hole through a variety of techniques. This improved location of the black hole was then named Sagittarius A* to separate it from the larger region of Sagittarius A.