But, I do think that this depends on the nature of the joke. In the example given in the OP, using the phrase "That's spicy!" as an expression of surprise is vague enough that it could simply become a phrase. But if it was something much more specific then I think it might emphasise the 'falseness'. If, for example, the OP and her friend had a phrase like "wait a minute... this isn't water, it's hot sauce!", then I think it would be slightly more off-putting, because it's more a phrase of reminiscence than an inside joke, and by using it, the second friend would almost be pretending they were there for the initial incident.
I think this is an interesting distinction. Let's say Sally and Sue are sisters, and Mary is their brother's fiancee. They want her to feel welcomed into the family. One day, in front of Mary, Sally and Sue use the phrase, "wait a minute... this isn't water, it's hot sauce!!!" and then start laughing. Mary's confused, because there's no water or hot sauce around. Again, in this scenario they want
her to feel welcome, so Sally and Sue happily explain the incident from a few years ago that inspired the phrase. Now Mary knows what they mean by it, and where it comes from.
I still think it would be weird for Mary to actually use the phrase herself, because it suggests she was present at an event that happened before her time. It's like she's artificially trying to build history with the group. If she does this kind of thing often, it could be viewed as a bit insecure and desperate, like she thinks she won't be able to form new
memories with the group so she's trying to insert herself into old ones. If I was Sally or Sue, I wouldn't be thinking so much that Mary is rude, as she is insecure, and if I generally liked her and wanted her to feel welcome, I would try doing other things to make her feel included and make new memories with her, so she doesn't feel the need to rather obviously co-opt situations she wasn't in.
Now, if this entire phrase has become a catchphrase of the whole family, and is already being used by people who weren't there, it would be rude to make Mary feel awkward for using it herself.