The clues we have to the shape of the Milky Way are:
1) When you look toward the galactic center with your eye, you see a long, thin strip. This suggests a disk seen edge-on, rather than a ellipsoid or another shape. We can also detect the bulge at the center. Since we see spiral galaxies which are disks with central bulges, this is a bit of a tipoff.
2) When we measure velocities of stars and gas in our galaxy, we see an overall rotational motion greater than random motions. This is another characteristic of a spiral.
3) The gas fraction, color, and dust content of our galaxy are spiral-like.
So, overall, it's a pretty convincing argument. Of course, we have to assume our galaxy is not completely unlike the other galaxies we see--but I suppose once a civilization has accepted that it does not occupy any special place in the universe, arguments about similarity seem sensible.