I read the following argument showing that not every language is described by a grammar:

For a fixed alphabet $ \Sigma$ and variables $ V$ there are uncountable many languages over $ \Sigma$ since the power set of $ \Sigma^\ast$ is uncountable. But grammars are finite objects by construction and thus there are only countably many grammars. In total, there can only be countably many languages described by grammars. Hence, there are this uncountably many languages that cannot be described by grammars.

I understand the idea of the argument, however i am not convinced by how they show that there are only countably many grammars. What do they mean by "finite objects"? Couldn’t one just take the set $ \{ (V,\Sigma, P, S) | \; P\subseteq (V\cup \Sigma)^+ \times (V\cup \Sigma)^\ast,\; S\in V\}$ , which is clearly uncountable, to get uncountably many grammars? Or do the languages that they generate fall together so often that we only get countably many languages generated by grammars in the end?