I concur with sdr in the comments: The best interpretation of your friend's statement is the conjecture that there is a bijection between irreducible motives of rank n and cuspidal algebraic representations of $GL_n$, characterized by each motive being sent to an automorphic form with the same $L$-function. This is one of several statements that is sometimes called "Langlands reciprocity".
We can't use algebraic varieties instead of motives for multiple reasons. One can see this already in the statement of modularity, which is a bijection from elliptic curves up to isogeny to modular forms, i.e. not a bijection from isomorphism classes of algebraic varieties but rather objects in some other categories.
We could use Galois representations instead of motives here. This would have several advantages, chiefly that there exist multiple potential definitions of the category of motives, none of which are known to have all the desired properties, but one definition of the category of Galois representations (with $\overline{\mathbb Q}_\ell$ coefficients) which has most of these desired properties. One has to put additional conditions on the Galois representations following the Fontaine-Mazur conjecture - the magic words are "finitely ramified and de Rham". One disadvantage is that this is further from the category of algebraic varieties and thus further from the original statement.
We could try to remove the "irreducible" restriction, but this isn't a good idea: "irreducible" matches up with "cuspidal" on the automorphic side, so irreducible motives correspond to cusp forms, irreducible motives can be put together to make general motives, and irreducible automorphic representations can be put together (via Eisenstein series) to make general automorphic representations, but the way they're "put together" doesn't give a bijection - there's a notion of a nontrivial extension of two motives, like a nontrivial extension of two representations, but nothing comparable on the automorphic side. (Information about the extensions does appear automorphically, but in different ways).
On the automorphic side, the restriction "algebraic" can't be removed, as non-algebraic representations don't correspond to anything in algebraic geometry.
One can replace $GL_n$ with a general group $G$, but this is not very helpful: These don't correspond to any new motives, but only to motives with $\hat{G}$-structure (e.g. a motive with $SL_n$ structure is a rank $n$ motive with trivial determinant - this isn't some incredibly exciting new structure). We lose a lot, in that the correspondence is no longer a bijection.
