Do milky way stars line up linearly in the sky the way solar system planets do?

This might be a dumb question, but it just occurred to me that that the milky way galaxy is a flat disc in space, just like the solar system, albeit at a different angle of orientation. So when we look at all planets, they are all on the ecliptic. Likewise, is there some similar "milky way ecliptic" along which all stars in the galaxy line up? If not, why not?