It would be nice if I would not have to set up all my search engines and their short codes with every new installation but could sync them just like my add-ons and history.
Don't get your hopes up, this was first reported 14 years ago.
In the meantime, there is a workaround for this that's easy enough [source]:
Assuming copying from one PC to another
Note: when I reopened FF after doing this my default search engine had been changed, so you may need to manually reset the default.