This idea of data being statistically significant has come up, but that's also easy to replicate with random data once you know the distributions of the data. In practice, those distributions rarely change, especially around demographic data. However, I don't think I've seen a case where this has been a problem. I'd be interested to learn about one.
Even if you know the distribution of the data (which imo can be fairly difficult) replicating that can also be tricky. If you know that a gender column is 30-70 male - female, how do you create 30% male names? How about the female names? Are they the same name or do you repeat names? Does it matter? In some cases it does and in others it doesn't.
What we've seen is that it's really use-case specific and there are some tools that can help but there isn't a complete tool set. That's what we're trying to build over time.
Not only that, but a staging environment with prod-like data is also useful for running load, performance and E2E tests without actually impacting production servers or touching production data. In all of these cases, anonymizing production data is important as you want to minimize the risk of data leaks, while also avoiding issues that can happen when testing against real production data (e.g. sending emails to actual customers).