I think what JRE is trying to say is that the stories have something more to them than mere metaphors. They have emotional connection. There's a good scene for the benefit of the viewers (emphasis mine)

DATA: The Tamarian ego structure does not seem to allow what we normally think of as self-identity. Their ability to abstract is highly unusual. They seem to communicate through narrative imagery by reference to the individuals and places which appear in their mytho-historical accounts.
TROI: It's as if I were to say to you, Juliet on her balcony.
CRUSHER: An image of romance.
TROI: Exactly. Imagery is everything to the Tamarians. It embodies their emotional states, their very thought processes. It's how they communicate, and it's how they think.
RIKER: If we know how they think, shouldn't we be able to get something across to them?
DATA: No, sir. The situation is analogous to understanding the grammar of a language but none of the vocabulary.
CRUSHER: If I didn't know who Juliet was or what she was doing on that balcony, the image alone wouldn't have any meaning.
TROI: That's correct. For instance, we know that Darmok was a great hero, a hunter, and that Tanagra was an island, but that's it. Without the details, there's no understanding.
DATA: It is necessary for us to learn the narrative from which the Tamarians drawing their imagery. Given our current relations, that does not appear likely.

JRE gave the example of "Runs like a red moped." That's a metaphor, but it's also a mere colloquialism. What the Tamarians are doing isn't using mere metaphors (where I say one thing that means another), but tying into the emotional connection from the story itself.

The Tamarian needs that emotional connection (which is what a mere metaphor cannot do). Hence the analogy of "Romeo and Juliet" that Troi mentions. Juliet is pining for her lover on the balcony. It connotes romance and/or love. If I were to say "New York, when the towers fell." it would connote distress and/or horror, but only if you know the full context beyond it being a mere historical event. These are far more than simple metaphors, unfortunately. The emotional context makes it impossible to discern the meaning of what they're saying by merely understanding the words and what they mean.

What the Tamarian captain wanted to fight the beast together, and thus (with a common foe) forge a new story with an emotional connection. That's why his crew fought so hard to keep the Enterprise from pulling Picard back from the surface. They knew that process would take time. You can start to see that when Picard finally connects with the Tamarian captain. Once they have forged a bond, Dathon (the Tamarian captain) explains the story of "Darmok and Jalad at Tanagra", and thus Picard is able to emotionally connect his recent experience with the Tamarian story (emphasis mine)

DATHON: Darmok and Jalad at Tanagra.
PICARD: Our situation is similar to theirs. I understand that. But I need to know more. You must tell me more about Darmok and Jalad. Tell me. You used the words, 'Temba, his arms wide' when you gave me the knife and the fire. Could that mean give? Temba, his arms wide. Darmok. Give me more about Darmok.
DATHON: Darmok on the ocean.
PICARD: Darmok. (draws on the soil) The ocean. Darmok on the ocean. A metaphor? For being alone? Isolated? Darmok on the ocean.
(Dathon writhes in pain.)
PICARD: Are you all right?
DATHON: Kiazi's children, their faces wet.
PICARD: Temba, his arms open. Give me more about Darmok on the ocean.
DATHON: Tanagra on the ocean. Darmok at Tanagra.
PICARD: At Tanagra. A country? Tanagra on the ocean. An island. Temba, his arms wide.
DATHON: Jalad on the ocean. Jalad at Tanagra.
PICARD: Jalad at Tanagra. He went to the same island as Darmok. Darmok and Jalad Tanagra.
DATHON: The beast at Tanagra.
PICARD: The beast? There was a creature at Tanagra? Darmok and Jalad, the beast of Tanagra. They arrived separately. They struggled together against a common foe, the beast at Tanagra. Darmok and Jalad at Tanagra.
DATHON: Darmok and Jalad on the ocean.
PICARD: They left together. Darmok and Jalad on the ocean.
DATHON: The ocean. (another spasm) Zinda! His face black, his eyes red. Callimas at Bahar.
PICARD: You hoped this would happen, didn't you? You knew there was a dangerous creature on this planet and you knew from the tale of Darmok that a danger shared might sometimes bring two people together. Darmok and Jalad at Tanagra. You and me, here, at El-Adrel.

Picard then tells him the story of Gilgamesh, a human story with the same type of emotional connection. It tells Dathon he has succeeded. When Picard has to stop the fighting between his ship and Tamarians, you'll note Picard has to indicate he has understood that connection (emphasis mine)

PICARD: Hail the Tamarian ship.
WORF: Aye, Captain.
TAMARIAN [on viewscreen]: Zinda! His face black, his eyes red
PICARD: Temarc! The river Temarc in winter.
TAMARIAN [on viewscreen]: Darmok?
PICARD: And Jalad. At Tanagra. Darmok and Jalad on the ocean.
TAMARIAN [on viewscreen]: Sokath, his eyes open!
PICARD: The beast at Tanagra. Uzani, his army. Shaka when the walls fell.
(Picard holds up Dathon's journal, and the Tamarians beam it away.)
TAMARIAN [on viewscreen]: Picard and Dathon at El-Adrel. Mirab, with sails unfurled.
PICARD: (holds out the dagger) Temba, his arms open.
TAMARIAN [on viewscreen]: Temba at rest. PICARD: Thank you.

Picard uses the correct emotional stories to communicate. The Tamarians respond by declaring that Picard's adventure on the surface is now a new emotional story for them.