I started to write this as a comment to the original question, but it became too long. It is not disjoint from the previous expert responses, but it emphasizes a particular viewpoint.

One can (and one should) shift normalize any automorphic (in particular any modular) $L$-function so that the functional equation relates $s$ to $1-s$. Then the gamma factors identify the archimedean component of the underlying automorphic form much like the Euler factors identify the non-archimedean components. In particular, if the $L$-function of a holomorphic cusp form is so normalized (i.e. $s$ is related to $1-s$), then the gamma factors determine the weight (and vice versa). I say gamma factors, because the usual single gamma factor can be factored into two gamma factors by the doubling formula for the gamma function (for a modular $L$-function the "true gamma factors" form a pair).

So you can ask your question as follows. If we shift normalize the $L$-function of an elliptic curve so that the functional equation relates $s$ to $1-s$, then why are the gamma factors always the same? In order to ask this question you need to assume already that the $L$-function obeys a rather specific functional equation with gamma factors, and then the question inquires what the gamma factors can be.

It is possible that the fairest answer to this question (i.e. your question) is as follows:

It was an experimental fact that the gamma factors are always the same, hence the precise form of the modularity conjecture was formulated, which then turned out to be trueright, namely it was proved by great efforts of great mathematicians.

Once again, normalization of all automorphic $L$-functions is key in this discussion, it should not be underestimated. It is not the usual normalization favored by algebraic people.

Added: In a leisurely style one could say the following. The first miracle is that the $L$-function of an elliptic curve is entire and satisfies some functional equation. The second miracle is that the $L$-function is automorphic, as suggested by the functional equation. More specifically, it looks like a $\mathrm{GL}_2$ automorphic $L$-function. Not only it is $\mathrm{GL}_2$, but it comes from a very specific modular form, namely a holomorphic form, let's call this the third miracle. Then, as a final miracle, this holomorphic form is always of weight 2 whose level can also be specified in terms of the elliptic curve.

1

I started to write this as a comment to the original question, but it became too long. It is not disjoint from the previous expert responses, but it emphasizes a particular viewpoint.

One can (and one should) shift normalize any automorphic (in particular any modular) $L$-function so that the functional equation relates $s$ to $1-s$. Then the gamma factors identify the archimedean component of the underlying automorphic form much like the Euler factors identify the non-archimedean components. In particular, if the $L$-function of a holomorphic cusp form is so normalized (i.e. $s$ is related to $1-s$), then the gamma factors determine the weight (and vice versa). I say gamma factors, because the usual single gamma factor can be factored into two gamma factors by the doubling formula for the gamma function (for a modular $L$-function the "true gamma factors" form a pair).

So you can ask your question as follows. If we shift normalize the $L$-function of an elliptic curve so that the functional equation relates $s$ to $1-s$, then why are the gamma factors always the same? In order to ask this question you need to assume already that the $L$-function obeys a rather specific functional equation with gamma factors, and then the question inquires what the gamma factors can be.

It is possible that the fairest answer to this question (i.e. your question) is as follows:

It was an experimental fact that the gamma factors are always the same, hence the precise form of the modularity conjecture was formulated, which then turned out to be true, namely it was proved by great efforts of great mathematicians.

Once again, normalization of all automorphic $L$-functions is key in this discussion, it should not be underestimated. It is not the usual normalization favored by algebraic people.