Just like there isn’t a universal ‘good’ saddle choice for everyone, the ideal shoe for different situations will be different from person to person. Generally speaking, a well made, perhaps custom or variable sized off the shelf, insole will make more difference than shoe brand. Different brands have different fit characteristics as far as toe box width, heel cup width/stiffness/height, overall midfoot height, etc. Cleat placement and insole stiffness/mid foot posting/calcaneal varus-valgus support will have a bigger effect than shoe brand on knee tracking.
As a general rule, you want even weight distribution between the lateral and medial condyles of the femur and their respective menisci and the tibia. It’s also good to minimize rotational forces at the tibia throughout the pedal stroke. There might be exceptions here, if you have symptomatic meniscus tears or spots of condylar pathology (various forms of arthritis, softening, etc). Assuming your fitter was aware and educated about your specific knee anatomy and issues, he may have made the recommendation towards Specialized brand shoes to help even out that force. If your foot doesn’t jive well with a Specialized shape, know that a similar varus effect can be had in any shoe (if that shoe doesn’t already have varus posting) by putting a very small (1-3mm) shim under the length of the medial ~1/3 of the insole, from just behind the ball of the foot to the front part of the heel. This is a crude solution, and I usually like to find more elegant (comfortable/durable) fixes for long term use, but it’s a great way to sample various amounts of calcaneal varus/valgus without having to continually buy new shoes.