How much is a $ of earnings worth now, how much was it worth then? Lower interest rates justify higher stock multiples than higher interest rates. Until not too long ago we had historically low interest rates, justifying comparatively high multiples. Only right now valuations look high when taking this all-important factor into account. ...
Shouldn't comparing "today" with "then" limit the "then" to years with comparable interest rates?

Nope. Not if you're interested in future stock returns, as I am.

I think there is a very important distinction to make.

Lower prevailing interest rates explain higher stock multiples. It's what historically happens, so that generally seems to be the reason it happens.

But lower interest rates don't justify higher stock multiples. The future earnings of those equities, and consequently the medium-to-long run returns you can expect from them, are unchanged. So a higher price just means a lower future return. The stocks aren't worth any more than they were at higher interest rates*.

That's solid theoretically, and solid empirically: stock returns from purchases made when interest rates are low and valuations are high are invariably poor. It's the starting valuation multiple on the stocks that determines the future returns, not the prevailing nominal interest rates at the time of purchase.

This old paper sets it out very succinctly.
aqr.com - Fight the Fed Model

One snippet:
"A simple analogy might be helpful. Say you can successfully show that teenagers usually drive recklessly after they have been drinking. This is potentially useful to know. But, it does not mean that when you observe them drinking, you should then blithely recommend reckless driving to them, simply because that is what usually occurs next. Similarly, the fact that investors drunk on low interest rates usually pay a recklessly high P/E for the stock market...does not make such a purchase a good idea, or imply that pundits should recommend this typical behaviour.
...
In fact, 1999-2000 is a nice example of the difference between describing how P/Es are set versus justifying them. When the fitted series peaked in the 40s in 1999, it was not saying that this P/E is rational for the S&P 500 (it was not). It was saying that, assuming investors act the way they have in the past, and given how low equity volatility had been versus bond volatility, and how low interest rates were, such an irrational high P/E was to be expected. The Fed model, alone or modified for volatility, offers no solace to those buying the S&P 500 at a P/E of 44, but it does explain what tricked them into doing so."

Jim

* higher real interest costs for indebted companies is a tiny quibble by comparison to the large valuation multiple swings, so it can be largely ignored for this discussion. It happens to some firms, and the reverse happens for others like Berkshire.