Is it ethical to claim experience with big data when the data isn't a part of the new advertising/social media/retail fad?
Data can be big in various ways. It can be large N (observations) and small P (variables per observation), like access logs for example. It can be large P and small N, like in biostatistics where you have tens of thousands of gene expressions from tens or hundreds of people. Or it can be both N and P, like in Facebook data. Data can also be made big, by normalizing every single detail of it, meaning that you'll be able to make a massive relational database schema that actually represents simple data.

All kinds of big data needs different approaches and companies mostly care about their own version of big data. That's why the answer isn't black and white.

To answer the question you posed in title then I believe it's ethical to claim experience with big data as long as you can put it into context of the business or domain where you encountered it and can describe some situations where a different set of tools or skills was needed than is used for smaller toy examples.