Was it a joke?
It could be joke, or it could be serious statement. Elasticities, especially their empirical estimation, became quite important in the field nowadays. Google Scholar shows over 24000 results for search on papers that estimate elasticity even when we restrict range just to 2010-present.
However, in 1947 people did not even had computers (they had people called computers but they did not had electronic computers), and it is plausible that he thought of elasticity as something that will never be measurable so doing calculations with it is just an good mental exercise for students but has no real world relevance.
However, he could be just making joke, since even purely theoretically elasticity can be interesting concept. Reviews of his contributions/academic biographies (e.g. Backhouse 2015) do not mention any dismissing view of elasticities so even if he was serious he did not spend much time advocating/developing this critique of elasticity
Did something happen with elasticities over the course of history?
Theoretically the concept did not change too much from the times of Samuelson. The major development in last 50-80 years when it comes to elasticities is that we learned how to empirically estimate them from data with reasonable accuracy.
Due to endogeneity issues, getting serious non-biased estimates of let's say price elasticity of demand, became possible only quite recently in late 90's and 2000s. Outside academia, in business environment, such techniques are just being adopted as we speak (since elasticity of demand is quite important when it comes to pricing decision), or at least so I was told in one of my econometric graduate courses.
Is there some truth to it?
Perhaps there was some truth to it in the past. Elasticities are still used as a practice problems in textbook, so they are definitely good mental exercises.
However, they became extremely important in quantitative economic analysis. For example, tax elasticities are absolutely crucial for optimal income taxation (see Saez 2001), price elasticities of demand are important for optimal pricing or advertising decisions, elasticities are also used to investigate terms of trade or to analyse consumption and saving decisions. Elasticity nowadays is definitely an important component of policy decision analysis, so the statement would not hold true nowadays.
Why does he call it dimensionless?
Dimensionless in sciences is synonym for unitless. Kg, cm, \$ are considered dimensions. So for example, US GDP's dimension is \$. Elasticity is unitless/dimensionless. It is just some real number that represents the how much % change in one variable changes another one variable, but it has no unit of measurement.