I don't know if my answer will help or not, but I will give it a go and let you be the judge. First, my Sony lens will exhibit a little side to side movement when I push on the side of the lens barrel. The degree of movement is slight and I cannot see visible movement. I primarily sense the play when the lens reaches the end of the inner mount and it is more a tactile sensation. I might hear a little click as the lens reaches the end of its travel. If I attempt to tilt the lens away from normal (i.e. displace the optical axis away from the perpendicular), I cannot do so. The degree of play is pretty much what I experienced when owning a Nikon F and Pentax LX film cameras.
Some of the lateral play is expected as the lens is held against the lens flange with springs. These have to be tensioned to allow the lens to be rotated when it is necessary to remove the lens from the camera. To allow for ease of removal they cannot be tightened too tight. However, if the lens is so lose that you can tilt the optical axis from the vertical then you have a problem. But a little lateral movement (side to side) is to be expected and my guess is that you are OK.
I used to sell cameras (film) and the only camera mounts that I found would exhibit no play and which would tighten rigidly was the screw mount used by Pentax and Practical and the breech lock mechanism of the Canon FD cameras. However, the Minolta, Nikon, Pentax K series cameras, and the Olympus OM-1 all had some of this lateral play and this was not a problem. Some of my customers told me that the springs can lose tension from repeated removal and remounting of lens, but I never saw one exhibit this symptom.
When using adapters with the Sony's mirrorless cameras, I do not use the inexpensive $20 units on eBay. The quality variance is too high and for my work with microscopes, I cannot afford to have any misalignment along the optical axis. Some of these low cost adapters provided a minute tilt and this would show up as a variation in focus--one side of the image would be blurred, as one moved to the center the image would sharpen, and finally when one looked at the other side of the image frame, it would be blurred again. With thick sections, 10-20 microns, most histologists will not see this; however, if you work thin sections of 0.5 micron and you are pushing the resolution limit of the microscope, this becomes a problem. The expensive adapters from Voigtlander and Tamron have not shown this characteristic and I only bought the inexpensive adapters so i could run comparisons between them and the more expensive adapter. I noticed this when using the APS-C sensors and it is perhaps unsurprising it is worse with the full frame cameras. I might add that the low cost adapters showed more of the lateral play, I described above. The play in the expensive adapters was more akin to a Sony made E-mount lens.
I hope this helps and it is my sincere hope your camera is OK.