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Both the pink ribbon Zack gives to Aerith and the flower Aerith gives to Cloud are gifts made in order to thank Aerith/Cloud for their help...
...and after Aerith/Cloud answer perplexed, Zack/Aerith offer it as a memento of their first encounter.
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[Another post about Aerith's ribbon]
It has been rumored by some fans that the Crisis Core scene where Zack gave Aerith the pink ribbon is no more canon, because kid Aerith has a pink ribbon too.
I don't mean to sound pretentious but...they're evidently different. Kid Aerith's ribbon has four ends, adult Aerith's ribbon has just two.
Since the original FFVII kid Aerith's had two outfits. She wore the blue outfit in the first part of Elmyra's flashback and the orange one when Tseng tried to convince her to return to Shinra.
In the Remake she wears the orange outfit only in the train graveyard vision and the blue one in all of Elmyra's flashback, from the moment she found her at the station up to when Tseng showed up. It's definitely not very evident because of the sepia tone filter, but we can compare her dress to the concept art and to the orange dress: she has no ribbon on the chest but wears a pinafore dress and a light shirt with bell sleeves.
This means that in the sepia toned scene she was wearing the green ribbon, as we can see from the concept art.
The scene at the train graveyard bears many similarities with the situation described in Picturing the Past, so she was wearing the orange outfit before escaping Shinra HQ.
When Elmyra found her and brought her in Sector 5 she had the green ribbon, not the pink one.
Someone tried to say Aerith stated the ribbon was a gift from her mother when she met Cloud at the church...
Nice try, but she was talking about White Materia. (Leaving a couple of links in case anybody wanted to check: OG, Remake)
So, not only there's no evidence that the pink ribbon is no more Zack's gift but that scene influenced also an important moment of the Remake: Aerith and Cloud's first encounter.
And in both cases the gift was made in order to thank the other for their help
CC: To show you my gratitude for that "hello" that woke me up.
Remake: You know, for scaring those things away.
Ehhh...can I say what a coincidence again?