I own an A7R3 and both of the lenses in question. I haven't noticed the behavior you are describing.
Several things to note.
@Clix Pix is correct in stating that you do not need to worry about AF fine tuning on the A7R3--one of the advantages of mirrorless is that focus is coming from the sensor and lenses do not need to be calibrated. However it is still possible for lenses to be "off" regarding performance--you can still get a bad sample and Sony is kind of bad in that regard.
Native ISO for the A7R3 is 100. This should be the lowest ISO that you ever use. Several of your examples are at ISO 80 or 50. Shooting at a lower ISO than 100 can introduce artifacts and it should be avoided.
Sony RAW files can have noise even at the base ISO of 100. It's not obvious in every shot, but it can be there for reasons not clear to me. See for example:
https://diglloyd.com/prem/prot/ML/SonyFullFrame/SonyA7R_III-noise-SprinterVan.html or
https://www.dpreview.com/forums/post/56280397.
Both of these are camera issues however and should not vary when comparing one lens to another.
You don't seem to have performed direct comparisons between the two lenses in a controlled manner. I would suggest this as your next step.
The subject needs to be the same. The ambient light and your exposure settings need to be exactly the same. Since the 55mm lens is a longer focal length than the 50mm lens, the compositions will be challenging to get exactly the same. It probably won't matter though for the purposes of the test. Having the same light on the subject and matching exposure settings *will* matter though.
Ideally you will want both exposures to be "to the right" on the camera histograms, meaning that they will be slightly overexposed without blowing out highlights. This isn't critical however since you will be comparing the two lenses and as long as the light and the exposures are held constant, the comparison should be valid. But as
@mollyc said above, noise lives in the shadows so it is better (in general) to avoid underexposure (with any shooting, not just with this test).
The focus point should be the same with each lens--ideally obtained by zooming in with Live View. Or zooming in using the built-in EVF. Focus should be in the center of the frame and this should be the area you crop to in post.
I would perform the test at relatively open apertures (say f/2) and also somewhat stopped down (say f/8). Ideally would do the comparison on a tripod where ISO 100 can be used and the shutter speed becomes irrelevant because you won't see motion blur (because the camera is on a tripod). Alternatively shoot handheld with a shutter speed that is fast enough to mask any camera shake (like no slower than 1/125th sec, just to be safe). Yes, IBIS is awesome but for this test you are trying to compare the two lenses and you want to control for all variables as much as possible.
Shoot with both lenses, again making everything as constant as you can between the shots with the two lenses. Shutter speed must stay the same. Aperture must stay the same. ISO must stay the same. Focus point must stay the same. The ambient light must stay the same (so either shoot indoors with the same light or outdoors only on a day without clouds or with uniform overcast cloud cover).
In post, look at both images at 100%. Do *not* look at higher magnifications (as you seem to have done in your crops in this thread) as they are not accurate. Because the focal length isn't the same between the two lenses, it will be tempting to crop to 100% on the 55mm lens and then crop to >100% on the 50mm lens so the crops look equal. The 50mm lens will look worse (and should look worse) because you are cropping it beyond the sensor resolution.
The results should be that the two lenses are fairly close to each other. They are both very good lenses. I prefer the 50mm in general because it is 2/3 stops faster and while the difference between 50mm and 55mm is small, my "ideal" walk about or general purpose focal length is closer to 40mm and 55mm is just too long for me (which is one reason among many why my current favorite general purpose prime is the Sigma 40mm f/1.4--but that's a topic for another day!).
If under controlled testing, your Sony 50mm f/1.4 is significantly worse in IQ compared to your Zeiss 55mm f/1.8 then the problem may very well be that you got a bad copy of the lens. Sony quality control regarding lenses isn't ideal, so this is a very real possibility. I'd consider returning the lens and seeing if a different copy performs better.