The type of photography that you are planning to use is an important decision factor. Here we have evaluated the Sigma 90mm F2.8 DG DN C and the Sony FE 135mm F1.8 GM for their suitability to various photography scenarios:
Size and Weight Comparison
The size of a lens is a crucial factor to consider when comparing two lenses. Sony FE 135mm F1.8 GM is the longer of the two lenses at 127mm. The Sigma 90mm F2.8 DG DN C with a length of 60mm, is 67mm shorter. Besides being longer, the Sony FE 135mm F1.8 GM also has a larger diameter of 90mm compared to the Sigma 90mm F2.8 DG DN C's 64mm diameter.

Sony FE 135mm F1.8 GM vs Sigma 90mm F2.8 DG DN C Lenses Size Comparison
The weight of a lens is equally significant as its external dimensions, particularly if you intend to handhold your camera and lens combination for extended periods. Sigma 90mm F2.8 DG DN C weighs 295g, which means it is 655g (68%) lighter than the Sony FE 135mm F1.8 GM which has a weight of 950g.
Below you can find the dimension and weight table of the Telephoto Prime Lenses in Sony E Mount for size comparison purposes.
Telephoto Lenses in Sony E Mount for Size Comparison
Filter Threads
The Sigma 90mm F2.8 DG DN C has a filter size of 55mm whereas the Sony FE 135mm F1.8 GM has a 82mm diameter. Larger filters are generally more expensive than the smaller ones given all the other features are equal.
Below are links to the 55mm filters that we recommend you to consider for the Sigma 90mm F2.8 DG DN C:
And these are the 82mm filters that we recommend for the Sony FE 135mm F1.8 GM:
Lens Mounts
Both the Sigma 90mm F2.8 DG DN C and the Sony FE 135mm F1.8 GM has the same Sony E lens mount. Some of the latest released cameras that are compatible with the Sony E lenses are Sony FX5, Sony A7R VI and Sony A7 V.
Sigma 90mm F2.8 DG DN C is also available in the Leica L mount.
Focal Range
Sigma 90mm F2.8 DG DN C is a prime lens with fixed focal lenght of 90mm.
On the other hand, the Sony FE 135mm F1.8 GM is a prime lens with fixed focal lenght of 135mm. When it is mounted on an APS-C sensor camera with 1.5x crop, it provides a 35mm (FF) equivalent of 202.5mm.