There is often a huge emphasis on analytical skills, instead of UX skills, when it comes to product management job descriptions. What role does UX play in product management?

It likely depends on the type of role you are entering into, the focus and phase of the product. It also will likely depend on whether the company has invested a lot in building out an existing UX team. I am biased because I have built out product design teams and therefore see true value in anyone who is working on product understanding experience fundamentals. If you have the best idea for a product but no one understands how to actually use it and get value, what use is the product to begin with. Instead of coming up with that great idea and then throwing it over the fence to you UX team, it would be better to approach your customer problem roundly and understand what you are hearing fully. 

I'm actually curious if this is widespread, as I've not seen in current or previous Product roles analytical skills being considered more valuable than UX... 

While analytical skills are important for looking at customer data in aggregate and prioritizing problems by impact, UX skills help PMs dig into the "why" and "how" our customers think and operate. It's a balance: We don't want PMs that "hide behind the numbers," nor do we want PMs that entirely base roadmaps on a handful of customer interviews / limited qualitative feedback.