I would hazard a guess that the latitude is off by 1 degree, and the coordinates may refer to Williamstown Observatory, because using 37 instead of 38 degrees, the coordinates would be S37° 52' 6.6", E144° 54' 49.5", a mere 200 meters away from the Williamstown Timeball Tower (37°52′00.6″S 144°54′45.7″E) (seen on the right in the picture below). This site had an observatory named Williamstown Observatory that was used until 1863 before the astronomical instruments were moved to Melbourne Observatory.
The observatory had transit instruments allowing to determine accurate timing by observing stars, similar to the Greenwich Observatory in the UK. It also served as a standard meridian for the early Geodetic Survey of Victoria (1858-1872)
Here are some interesting links for further reading on the Williamstown Observatory:
Indeed, the coordinates seem to fall a bit offshore when we use the modern WGS84. This isn't surprising, because back then they determined the geographic coordinates of Williamstown Observatory astronomically, which typically can differ from modern values by hundreds of meters, because of different sources of errors in the astronomical method. For example, the deflection of the vertical being around 7"S, 4"W (estimated from EGM2008 geoid), even with perfect instruments and chronometers, they would measure the coordinates to be 7"S, 4"W of the modern coordinate in WGS84.
Since Williamstown served as an important reference benchmark for laying out the Geodetic Survey of Victoria, the rest of the survey follows the then determined coordinate and is "off" by a certain amount when compared to WGS84. Also, the use of a different ellispoid (Clarke 1858) to compute coordinates away from Williamstown also contributes to differences in geographic coordinates when you compare to WGS84.
This old 1874 document seems to report the coordinates of Williamstown Observatory as S37° 52' 7", E144° 54' 42". I don't know if this is the reference coordinate they used, but if we knew them for sure, then it might be possible to calculate an approximate mathematical transformation from that datum to WGS84. Ideally, we would need a reference azimuth, or other reference points, to determine a transformation. There is a drawback to the mathematical approach though; positions derived from surveys back then weren't quite as consistent as today, so transformation errors are still to be expected, but for a relatively small region (eg a city), it could give an interesting approximation. Maybe a transformation already exists, but I haven't found any.
However, if the coordinates you are talking about were determined purely astronomically, then I'm afraid the margin of error of these instruments is too large to determine a consistent transformation.