How did the object CO-0.40-0.22 get its name, and how is it distinct from CO-0.40-0.22*?
An earlier paper on the object (Oka et al. (2015)) explains that CO-0.40-0.22 is a &quot;high-velocity compact cloud&quot;. The first discovery of such an object was two decades ago (Oka et al. (1998)), when CO 0.02-0.02 was found. The naming convention used for that object was
&quot;CO&quot; + Galactic longitude + Galactic latitude

For instance, CO 0.02-0.02 has latitude $0.02^{\circ}$ and longitude $-0.02^{\circ}$; CO-0.40-0.22 has latitude $-0.40^{\circ}$ and longitude $-0.22^{\circ}$. Naming based on coordinates is used for many other radio sources.
The * in the name of the continuum point source is, I believe, in continuation of the tradition set by the discovery of Sagittarius A*, the point source in Sagittarius A. It is often necessary to distinguish between the two. Quoting from Goss et al. (2003),

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. Brown (1982) proposed a model of Sgr A* consisting of twin precessing jets with a period of 2300 years. The model has not stood the test of time but the name immediately was accepted. As an example, the VLBI results discussed by Lo et al. (1985) uses the name Sgr A*; the review article by Lo (1987) also uses this nomenclature.
Bob Brown provides the following rationale for the name: “ 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.”

The difference between CO-0.40-0.22 and CO-0.40-0.22* is simple: The former is a high-velocity compact cloud, and the latter is a point radio source inside it, possibly an intermediate-mass black hole. The formation, evolution and interactions of the two are likely tied together, but, as with Sagittarius A and Sagittarius A*, they need to be distinguished.