It is the man who says, "That's the way with everything." There are a lot of "he saids" and "she saids" left out of the dialogue because Hemingway characteristically did not want to explain what should be obvious. The lines of dialogue in question are as follows"
"It tastes like licorice," the girl said and put the glass down.
"That's the way with everything."
"Yes," said the girl. "Everything tastes of licorice. Especially all the things you've waited so long for, like absinthe."
Obviously it is the man who says, "That's the way with everything."
They are both getting disenchanted with traveling around in Europe. "Distance lends enchantment"--but when things are seen close up they are often disappointing. She is becoming disenchanted with the man, and he is becoming disenchanted by the mess he has gotten himself into.
The girl wanted to try a drink called Anis del Toro. She was disappointed. The man, who is obviously older and more experienced, seems to have accepted the fact that life is full of disappointments. He already knows what Anis del Toro tastes like. But they are both thinking of life in general rather than about the tastes of various kinds of liquor. The girl is thinking about the embryo in her womb. She has probably dreamt of having a baby since she was a little girl, and now the harsh reality is entirely different.