There are quite a lot of categories here. I have not tried very many of these lenses. What you read here is just the summary of a lot of research. Here we go …
1. Kit lens
The two main kits lenses have the same focal length range (16-50mm) and the same aperture (f/3.5-5.6). The big differences are that the Sony has a power zoom function, is a lot lighter in weight and collapses down to a small size. Some would argue that there is also a big difference in image quality. A kit lens is never going to be of premium quality so the question with any kit lens is if the IQ is good enough for a beginner lens. Assuming that the Sony is of worse IQ than the Fuji, does the smaller size and power zoom make up for it? The answer will vary from person to person. If IQ is most important to you then Fuji probably wins here. If weight and compactness, then the Sony wins.
2. Standard zoom
Fuji has two options here and Sony has three or four depending on what you include. The first comparison is the ‘step-up’ zoom:
Fuji XF 18-55mm F2.8-4 R LM OIS
Sony Carl Zeiss Vario-Tessar T* E 16-70mm F4 ZA OSS
On paper, the Sony seems to have the edge here: it is wider at the wide end and longer at the long end. The Fuji goes to f/2.8 at the wide end but only f/4 at the long end, the same as the Sony. Both lenses are stabilized. The Sony is slighter lighter in weight.
That said, the Sony is substantially more expensive and there has always been a question over the image quality for that price, with vigorous debate on both sides.
B+H review ratings are very similar: an average of 4.6 for each lens. Reading the reviews there are a few negative comments on IQ for both, but less (I think) for the Fuji. On that basis, there is a simple trade off here between the greater range and some Zeiss magic of the Sony against a faster, cheaper and (slightly) sharper Fuji lens.
The other options are less comparable. I will not cover the FE lenses here.
Fuji has a premium option: the 16-55mm f/2.8 for $1,200 and weighing 655 grams but lacking stabilization. Sony does not really have an equivalent lens – if there was one it would have similar specifications. Compared to the Sony Zeiss 16-70mm, the Fuji is faster and sharper but lacks stabilization and is twice as heavy and somewhat more expensive.
I have to give the edge to Fuji here simply because there are two options: a credible mid-price zoom and a premium fast zoom. If the Fujis had the extra 20mm at the long end then it would be no contest, but they don’t.
3. Longer zoom
This is a duel between these two lenses:
Fuji XF 18-135mm F3.5-5.6 LM OIS WR
Sony E PZ 18-105mm F4 G OSS
The Fuji is longer, a little heavier and a lot more expensive. The Sony has an internal zoom, which is good for dust resistance, but the Fuji is weather sealed. The Fuji is 1 cm shorter when retracted. The power zoom of the Sony is aimed at video shooters: some stills shooters like it, others not so much.
Image quality is difficult to judge. The Sony has a lot of distortion, which some people think is a deal-breaker and others could not care less. Both lenses are reasonably sharp for lenses with a large focal range.
This is all going to come down to what you think of the power zoom (which resets to 18mm when you turn the camera off), if you think that the distortion is a big deal and if you think that the extra 30mm of the Fuji matters.
For me, the lower price of the Sony puts it ahead of the Fuji, but others will passionately go the other way.
4. Wide zoom
Fuji XF 10-24mm F4 R OIS
Sony E 10-18mm F4 OSS
These are both very highly regarded lenses. The difference is the extra focal range for the Fuji at $150 more than the Sony. There is not a lot else to say here. I think the edge must go to the Fuji here – the extra range is significant.
5. Telephoto budget zoom
Fuji XC 50-230mm F4.5-6.7 OIS
Sony E 55-210mm F4.5-6.3 OSS
These are also similar lenses. Cheap and slow. The Fuji wins for its slightly greater range.
6. Telephoto zooms
The lenses are difficult to compare here since they fall into somewhat different categories. At this length, we can consider the FE lenses since an equivalent hypothetical APS-C lens would not be much (if any) smaller.
Fuji XC 50-140mm F2.8 OIS
Fuji XF 55-200mm F3.5-4.8 R OIS
There are no Sony lenses in these categories, which is a major shortfall in my opinion, particularly the $700-ish 55-200mm. On the other hand, Fuji does not have anything which competes directly with these (also a major shortfall):
Sony FE 70-200mm F4 G OSS
Sony FE 70-300mm F4.5-5.6 G OSS
Finally, Fuji has the monster:
Fuji XF 100-400mm F4.5-5.6 R LM OIS WR
Its difficult to pick a winner here. Fuji’s premium lenses are very heavy and become niche options. The 50-140mm is a stellar lens, however. I think that the Sony FE lenses are a nice mix of quality and weight and are therefore more all-round useful but ultimately the Fuji 55-200mm tips it in favour of the Fuji system.
7. All-in-one zoom
Sony E 18-200mm F3.5-6.3 OSS
Fuji does not have an option here, while E mount has three (four if you count the FE 24-240mm). I don’t think this means much – Fuji has chosen not to enter this market and the Fuji 18-135mm can handle some of the demand for the all-in-one.
So, as a Sony shooter, I’m afraid that the bottom line here is a clear win to Fuji. Their lens selection is well thought out, with good options at a range of price points. Note that I have not in any way concluded that the Sony zooms are ‘junk’ or anything silly like that. The 10-18mm is a fine lens. The 18-105mm is a very good value lens. The 70-200mm and 70-300mm are excellent lenses which expose a gap in the Fuji line-up. Remember that there is not clear winner in terms of the Sony and Fuji primes.
For me, the major gap in the Sony zoom line-up is in the $700-$800 range where Sony oddly does not play: a standard zoom (16-50mm f/2.8-4) and a telephoto zoom (50-200mm f/3.5-5.6). This is not a wish list, just a statement of where the gap is.
I am doubtful that Sony will release any APS-C lens for at least a year, maybe two. If Sony ever releases a APS-C 16-50mm f/2.8 zoom it will not be less than $1,500.