iPhone 16 ProMax. I had the same problem: my stickers were not transferred from my iPhone Pro 14. I would try to create new stickers and they would disappear within a second.
1) I texted with Apple support and they told me they had no reports of this issue. Not true, because there are way too many people reporting this on multiple sites. But I digress. Apple support had me reset the Keyboard settings. This did not work. Next move was to reset Network Settings. That did not work.
In the mean time, I also found that my keyboard shortcuts for various cuss words were not transferred from my other iPhone. This also bugged me.
2) Tried a bunch of resets and other such futile tasks.
3) After 5 days of being annoyed because I love my stickers - say what you want about that - I read on a forum that the iPhone 16 promax was shipped with a preloaded version of iOS 18 and that is where the bug is. Per the site, I performed a full reset on my 16 promax. I restored my iPhone from the backup I performed just prior to the reset. 30 minutes later, stickers are working. My keyboard shortcuts and the words I so enjoy are also back. Sorry that is the only thing that worked.