Many years ago I read a short story about mankind searching the galaxy, hoping to find intelligent non-human life.
"Final Encounter" by Harry Harrison, first published in Galaxy Magazine, April 1964, available at the Internet Archive. Previously identified as the answer to this question.
After many millennia, on the opposite side of the galaxy, they think they have finally found an alien species, who have been undertaking the same quest. Later, DNA analysis reveals that the supposed alien race are actually descended from humanity, but evolved differently while going the opposite way around the galaxy.
"The galactic core explains partially what must have happened," Gulyas said. "It is thousands of light-years in diameter and over 10,000 degrees in temperature. We have explored its fringes. No ship could penetrate it or even approach too closely because of the dust clouds that surround it. So we have expanded outwards, slowly circling the rim of the galaxy, moving away from Earth. If we stopped to think about it we should have realized that mankind was moving the other way too, in the opoposite direction around the wheel."
"And sometime we would have to meet," Liem said. "Now I greet you, brothers. And I am sad, because I know what this means."
"We are alone," Hautamaki said, looking at the massed trillions of stars. "We have closed the circle and found only ourselves. The galaxy is ours, but we are alone." He turned about not realizing that Liem, the golden alien—the man—had turned at the same time in the same manner.
They initially conclude that mankind is alone in the universe, but then turn to look at the Andromeda galaxy, and say something like "No, we are alone in this galaxy".
"It is certain then," Hautamaki said, "we are alone in the galaxy."
"Alone in this galaxy."