Should product marketers focus in on being great at one specific niche (skill, industry, etc.) or be more of a generalist?

"It depends" is the most infuriating answer but this is an "it depends" moment. This is all dependent on where you are in your career, what you want in your career, and also what your propensity is. 

In terms of pure marketability, the broad strokes I'd make (and your mileage may vary) is: 

• Generalist >> you will have the most flexibility in the companies you work at but you may not rise as rapidly or command the most senior roles upfront. But you'll never be pigeonholed. This is really useful particularly when you start your career (first 5-10 years) before you figure out what you like.
• Super focused in an industry >> you will bounce around all of the companies in that industry and get to know them intimately, giving you a chance to be a real specialist and command niche roles. Just make sure the industry you're focused on is growing. 
• Super focused on a skill >> you will most likely end up in a larger company doing that skill. Smaller companies will probably not be able to afford you nor be able to employ you for that skill alone.
• Super focused on a type of company >> you can be a really good product marketer or marketing leader who works in particular type of company (ex. the first product marketing hire who builds the first team). This can be incredibly fun if you are interested in sticking to a type of organization and then moving from them as they grow. 

Just take a hard look at yourself and what you're curious about if you decide to specialize. The best part about product marketing is getting deep into what you're doing so make sure if you decide to specialize, that you really want to know the audience, the market, and the products. 

I think either is a valid path to a rewarding product marketing career but can depend on your own interest. If you're interested in the dynamics of different industries and you find that an inspiring aspect of being a product marketer, it makes sense to diversify your industry experience. If you find you're a rockstar at messaging and positioning, maybe you want to go deep into the skill and apply it to different companies in a similar space.

This is an interesting question. Sometimes it is hard to think of oneself as a product. 

What we're really trying to do here is to position ourselves for the maximum market impact. (I'm a PMM after all)

One of the concepts that stuck with me from Geoffrey Moore's book was the concept of the bowling alley. You only change either your market segment or your app (skills) at one time. (image below)

The most rational approach would be to find a large enough industry/domain,( I have just happened to work a lot in Customer Experience, Customer Service related domains selling into B2C companies), and then grow your skills continuously to improve your market power.  

Not evolving outside of (App1), i.e. niche skill is going to limit your market potential, so you better make sure your (Seg1) is large enough to sustain you. PMM is niche as it is, so I personally would not counsel folks to be stuck in a further niche.