The closest discussion I can find is Expanding abbreviated tag names so apologies if this is a dupe.
There is no very deep reason. In the post you linked to a list of very short tags was highlighted. I then went through the list and decided what to do, commenting:
I did something about all proposals. I am not sure I chose the best option each time but it's easy to change. Feel free to comment obviously. Maybe it's best to take the changed version as a new base line and to propose changes in the usual ways (e.g., post in tag management thread)
Sometimes I expanded the tag-name, in most cases keeping the abbreviation as a synonym, sometimes I kept the short tag and added the expansion as a synonym. My main guiding principle was my perception of how common the abbreviation was; I mean some abbreviations are way more common than the actual thing (e.g., CPU; or even laser, which is by now an actual word).
My reasoning at that point was that, as alluded to above, in my impression, PDE (as abbreviation) is widely used, people even used it commonly and naturally in spoken language. I am not sure this is true to the same extent for SDE.
That said, presented as you just did, I tend to agree that consistency of the tagging scheme should take precedence over the subjective reasoning, and certainly "partial differential equation" is also commonly used.
Thus, I propose to switch the direction of the synonym for pde so that all three have the full name as tag. I prefer this over all having the abbreviation as tag; I am not sure I would easily recognize sde in isolation even thought I do have (rather limited but still) knowledge of the subject.
If there is no major opposition within a couple of days, I will switch the direction of the synonym, which is a painless and reversible change. If I forget, please leave a comment on this post.