This question is probably really easy to someone that knows the theory of modular forms well, so I apologize if this is obvious. Suppose $E_1$ and $E_2$ are elliptic curves over $\mathbb{Q}$ and the Euler product expansion of the Hasse-Weil L-series of each matches at all primes of good reduction. Some references use the definition of modular to be that for all but finitely many primes, $L(E_1, s)=L(f,s)$ where $f$ is a (certain type of) modular form. Some references seem to say that modular means $L(E_1, s)=L(f, s)$ where it seems implied that all factors are the same.
My first question is whether or not there is a way to see directly from modularity properties (such as there being an analytic continuation of $L(E_1, s)$ that satisfies a functional equation) that there is some unique way to fill in the rest of the Euler factors. The second related, or possibly equivalent, question is whether or not you can tell that $L(E_1, s)$ and $L(E_2, s)$ have the same factors just from knowing the factors of good reduction.
Note: the motivation is from more general "Galois representations coming from geometry" where you don't know something nice about the relation between the varieties. For instance, if you prove this by first saying they must be isogenous it won't generalize.
Note 2: the reason I find this hard is that a priori you could have bad reduction of $E_1$ and $E_2$ at exactly two primes $p_1$ and $p_2$, and somehow it just works out that the factor of $L(E_1, s)$ at $p_1$ is exactly the factor at $L(E_2, s)$ at $p_2$ and vice-versa to give exactly the same L-function, but the factors are filled in differently (this can't happen in the elliptic curve case because we know the formula explicitly, but again I'm wondering if it can be seen without knowledge of this formula).
If you don't like this wording of the question using elliptic curves and then not being able to use that fact, the real question is if $X$ is a variety over $\mathbb{Q}$ and its middle $\ell$-adic cohomology is $2$-dimensional with the property that the L-series coming from the (contragredient) Galois representation matches the L-series of a normalize Hecke eigenform for all primes of good reduction of $X$ (an integral model was fixed), does this uniquely determine the other factors in the sense that if I had another modular variety of the same dimension with the same modular form attached I could conclude every factor of the L-series matches with every factor of $L(X,s)$?