Firstly, the question imply that (in English) the word the is not used by Rashi, but any terms we have are really carried over from Hebrew. In Hebrew, Rashi was often referred to and is still by some as רשי הקדוש and that would be the only way to say that properly in Hebrew. Thus, were that carried over in a widespread fashion, he would have been indeed called The Holy Rashi or The Saintly Rashi, while including the word the. So perhaps there is an additional question my he and others were not as specificity referred to like that. Another answer could be in essence of what the translation comes to be and perhaps the more common knowledge of Rashi’s abbreviated name. Because neither would make sense with the word the in the front.
The our Rabbi Moshe Ben Maimon
The our/my Rabbi Shlomo Yitzchaki
But your point is still present in Wikipedia.
In the Hebrew Wikipedia page for Rambam it starts his biography:
הרמב"ם נולד בקורדובה, שבאל-אנדלוס
By Rashi it starts:
רש"י נולד בעיר טרואה
It is important to note that this applies both by English (taken from Hebrew) and by Hebrew because these languages contain the word the and therefore can employ its use.
This is not present in other languages which do not employ the word the such as Russian and Chinese. (And other Slavic and Asian languages)
For example, copied from Rambam’s Wikipedia page in Russian:
Рамбам очень быстро стал играть важную роль...
Rambam (no the word existing or present) very quickly began to play an important role...
And by Rashi’s in Russian:
Раши родился в городе Труа,
Rashi (no the) was born in Truya (Troyes),