I have just started studying Macroeconomics and have noticed that as per the Classical economics, the real wage level will always remain same because any change in price level in the market is subsidized by an equivalent increase in the wage level in the market. However a look at the USA's hourly real wages over years (https://en.wikipedia.org/wiki/Real_wages) shows that the wages are anything but constant. Can classical economics explain the variation in the real wages or is this one of the shortcomings of the classical economics?
Classical economics is actually not taught at universities anymore (not since 19-20 century at least). You probably have in mind some Neoclassical model. I will presume that since you say you study macroeconomics and not only classical economics is now just part of history of economics, it for the most part did not dealt with macroeconomics.
You don’t mention in which model you observed this, but in general this has nothing to do with whether model is classical or neoclassical. For example, many business cycle models on purpose ‘turn off economic growth’ (which is what in long run drives also increases in real wage), implication of which is that also real wages remain same, and this holds for both Neoclassical or New Keynesian models.
This usually done on purpose so scientists and students can examine how economy behaves, for example during the business cycle, without having to track extra variables/equations and to also make math easier.
Going back to your question there are actually many Neoclassical models of economic growth where real wages increase. For example, in famous Solow-Swan technological growth leads to increases in productivity and real output per worker and hence consequently also real wages.
Neo-Classical economists assume that prices and nominal wages are flexible. They assume that since they are perfectly competitive markets, they will always clear. As you mentioned in your question, if there is an increase in price, there is an equivalent increase in nominal wages and voila! real wages remain constant.
This is far from the truth. The reason behind that is the fact that prices and nominal wages are ( what Keynes calls them) "sticky". A lot of times nominal wages are fixed by employment contracts, trade union activities, reluctance to accept cuts in nominal wages and efficiency wage theories. (Read: https://www.economicshelp.org/blog/154792/economics/sticky-wages/)
Similarly, an increase in nominal wages won't always make firms increase the prices of their goods. This is because prices are sticky too !. There are menu costs involved in changing prices. (Read:https://market.subwiki.org/wiki/Price_stickiness)
Since both our factors are sticky, the outcome i.e. Real Wages fluctuate.