How did the object CO-0.40-0.22 get its name, and how is it distinct from CO-0.40-0.22*? There seems to be three things described in the recent Nature Astronomy paper Millimetre-wave emission from an intermediate-mass black hole candidate in the Milky Way:

Molecular cloud CO–0.40–0.22 with an extremely broad velocity width, near the centre of our Milky Way galaxy.
a compact object with a mass of about $10^5 M⊙$ is lurking in this cloud.
a point-like continuum source called CO–0.40–0.22*

Question: What are the objects CO–0.40–0.22 and CO–0.40–0.22* and what is the difference between the two. How did the originally discovered CO–0.40–0.22 get its name?
The news of the paper is described in the Science item Long-rumored midsized black hole may be hiding out in the Milky Way.
There is a nice writeup in Gizmodo for lay scientists: Astronomers Think They Discovered The Missing Link in Black Hole Evolution
The paper is not paywalled and can be read here: https://www.nature.com/articles/s41550-017-0224-z.pdf

Abstract:
It is widely accepted that black holes with masses greater than a million solar masses (M⊙) lurk at the centres of massive galaxies. The origins of such ‘supermassive’ black holes(SMBHs) remain unknown, although those of stellar-mass black holes are well understood. One possible scenario is that intermediate-mass black holes (IMBHs), which are formed by the runaway coalescence of stars in young compact star clusters merge at the centre of a galaxy to form a SMBH. Although many candidates for IMBHs have been proposed, none is accepted as definitive. Recently, we discovered a peculiar molecular cloud, CO–0.40–0.22, with an extremely broad velocity width, near the centre of our Milky Way galaxy. Based on the careful analysis of gas kinematics, we concluded that a compact object with a mass of about 105 M⊙ is lurking in this cloud. Here we report the detection of a point-like continuum source as well as a compact gas clump near the
centre of CO–0.40–0.22. This point-like continuum source (CO–0.40–0.22*) has a wide-band spectrum consistent with 1/500 of the Galactic SMBH (Sgr A*) in luminosity. Numerical simulations around a point-like massive object reproduce the kinematics of dense molecular gas well, which suggests that CO–0.40–0.22* is one of the most promising candidates for an intermediate-mass black hole.
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.