This gift was programmed in the game to be shiny locked. However, due to a programming oversight,
it is possible to receive the Partner Cap Pikachu as shiny (standard odds),
if your Trainer Shiny Value (TSV) is not 1009. This is the only Pikachu Cap Variant that can be received shiny.
(Note: Still available shiny, up to update Version 1.2)
How the game mechanic worked, was that the game would generate a PID that is not shiny compared to your TID/SID.
After that generation, the game will proceed to overwrite the Pikachu's TID/SID with Ash's.
Hence, there is a chance that the PID generated for the Pikachu will be shiny against TID/SID of Ash, as the game doesn't check for that.
This was discovered by @wwwwwwzx when reviewing the game's code, who then went ahead to test and prove it with RNG.
A few other users also corroborated the data with their own findings.
The data below refers to the file itself, not the encounter as a whole.
Format Ver.1.0.6-1, Post Updated Date:20180207_2340