I think this has to do with concurent validity. A related concept is multi-trait multi-analysis analysis. Note that in my reply to the linked question, I rather say that "the traits are defined in a consistent and objective manner." This means that if we were to administrate two different questionnaires that purport to measure the same construct to the very same sample, we expect individual scores to be highly correlated, pending minor issues like measurement error which can be mitigated using a dedicated correction. The choice of the threshold (0.7 or 0.8) is up to you, of course, like any such criteria. They are often arbitrary since it all depends on the sample size, the length and reliability of the questionnaires, etc.
Setting aside the shared variance that can be explained from simply decomposing the correlation between two latent variables, I think it remains tricky to affirm that "constructs that share this much variance are similar and potentially interchangeable." First, you can reach a very high correlation with constructs that are not directly related, i.e. traits that can be found in isolation or combined (in health measurement, for instance, we used to think that measuring quality of life is essentially measuring depression); second, there's a gross approximation between constructs and scale scores (whether your scale effectively measure what it claims to has to do with construct validity), IMO. I believe these two points are in line with @Tim's comments.