“A dude doing undergraduage studies gets a degree in undergradute studies”
That’s likely the line of thought leading to the popularity of the phrase (well, in addition to another line of thought, to be delineated below), silly though it is. Silliness can arise from (marginally) respectable origin.
There isn’t really any such thing as “undergraduate studies.” You are the undergraduate, while the studies are whatever they are. Same goes for so-called “post-graduate studies.” You don’t study post-graduation. You study whatever it is you study while being a post-graduate.
These are merely convenient phrases, and harmless so far as they go. We don’t encounter a problem until we talk degrees. The simple truth is that you have to be graduated in order to have a degree. Therefore, there can be no such thing as an undergraduate degree. It makes no sense. The iron laws of logic forbid it.
Now, we arrive at the aforementioned alternate line leading to the popularity of the term. That is, the existence of the “post-graduate degree.” You might assume I am opposed to that phrase, and I might be, but not for logical reasons. That phrase is not very descriptive, and is in my opinion misused to bolster the high wall between people with degrees and those without. However, it is not silly. It makes sense. One can get a degree after they’ve been graduated. Therefore, the phrases “graduate degree” or “post-graduate degree” is acceptable. However, in no sense does its sense render the phrase “undergraduate degree” sensible.
The popularity of “post-graduate degree” may or may not (I’m not familiar enough with the history to declare) have led “undergraduate degree” to gain currency. It’s a regrettable eventuation.
And respectability can arise from silliness.