This doesn't prove anything, but it's interesting to consider at least: if you look at the list of the most common elements in the Milky Way[1], 7 of the top 10 are essential for life on Earth: Hydrogen (#1), Oxygen (#3), Carbon (#4), Iron (#6), Nitrogen (#7), Magnesium (#9), and Sulfur (#10). Silicon (#8) is also pretty much essential. The other 2 elements in the top 10, Helium (#2) and Neon (#5), are noble gases with low chemical reactivity.
So, what could that mean?
If you claim that those elements (or at least a subset) are essential for any kind of life, then by the anthropic principle, it's not surprising to see the distribution that we see in our home galaxy.
Alternatively, you could claim that life can evolve with all sorts of combinations of elements, but that it makes sense that it would evolve to use the most common elements in any given area of the Universe.
Either way, it seems like the fact that the life we know uses the most common elements in the galaxy is a good reason to expect non-Earth-based Milky Way life to be similar.
This is just a thought I've had in the past; I haven't done a whole ton of research into this correlation. And of course, there's a whole other set of arguments involving the chemical properties of elements, which I'm just skipping over in this comment.
[1] https://en.wikipedia.org/wiki/Abundance_of_the_chemical_elem...
EDIT FOR CLARITY: The sentence mentioning the anthropic principle used to read: "By the anthropic principle, you might say that those elements (or at least a subset) are essential for any kind of life, and given that life exists on Earth, it's not surprising to see the distribution that we see in our home galaxy."