How do you measure innovation?

We have KPIs for each stage of innovation.
1. Ideation: # of ideas, panel scores, # path-breakers
2. Incubation: Time, success with POC and business buy-in
3. Prioritisation and build: roadmap performance, time, effectiveness of buy/ build, idea drop off rate
4. Delivery: Benefits realised, user feedback, ROI, productisation, idea backlog

You can measure the outcomes, process and culture. Outcomes can be measured in the same way as any other initiatives - revenue uplift, risk reduction, cost efficiency, customer / employee experience, market share etc etc as long as these things are aligned with your strategic objectives. A good innovation process should have a defined pipeline of something like ideation, scoping, selection, pilot and scaling - the pipeline should be managed and discipline around the best ideas progressing through. Measuring innovation culture is tricker - how much is it in the company DNA - how many ideas are submitted per person, how many people are training in innovation toolsets and skills and how quickly are innovations realised? How do you benchmark yourself against the sector and wider market are also good questions to ask. 

Customer satisfaction can be used as a proxy for innovation.

If you are using "innovation" (it's not a verb) as a way to solve customer problems, it will be reflected back.

I measure innovation by seeing a need and developing or devising something to fill that need.  It's cliche but necessity is a spark of innovation.  That said, great ideas are only as good as the ones that can put those ideas into actionable goals and steps to produce outcomes.  

I am not a big fan of establishing any kind of formal innovation group, team etc. Why? Innovation can't be institutionalized, mandated, prescribed etc. in companies, industries or people who're not innovative to start with.

There needs to be however a way, a process to collect great ideas and innovative seeds from across the entire population in a company. Independent of an employee's rank, level, title, role etc. That does however anticipate a respective company culture (most of them fail there!), for the employees in a company to truly understand and identify themselves with the company's "purpose", vision, strategy (how many strategies out there are "readable", trigger purpose or meaning?) what their customers really need or want or what else there might be to tie more customers, create a new value proposition, optimize existing value propositions or eliminate unwanted/unpopular ones.
Bottom line to me: innovation shall be measured by how well, how creatively customer needs, business opportunities are being anticipated or responded to = revenue increase, revenue generation by means which go above and beyond of what's there today!