The Gemara in Kiddushin (38a) states that Moshe died on ז' אדר. The Gemara learns this out from the pesukim in וזאת הברכה and Yeshoshua. In וזאת הברכה it says that Bnei Yisrael mourned Moshe for 30 days. In Yehoshua it says that after they finished mourning for Moshe they prepared for 3 days and then crossed the ירדן. The Pasuk states that this was on the tenth of Nissan. If you count backwards 33 days you get to the 7th of Adar. The Gemara then comments that Moshe said on the day of his death that he is 120 years old היום meaning that he died on the day he was born because Hashem is ממלא the years of a Tzaddik to the day.
However, the Gemara in Sota (12b) has a machlokes when Moshe was born. ר' חנינא בר פפא says that the 3 months of hiding of Moshe ended on the 21st of Nisan, and therefore it had to be that Moshe was born on ז' אדר ראשון and the 3 months are most of אדר ראשון, אדר שני and most of Nissan.
According to ר' חנינא בר פפא when did Moshe die? The Gemara in Kiddushin's proof from the pesukim seems airtight and therefore it is hard to imagine how ר' חנינא בר פפא could argue on that. Therefore it must be that Moshe died on ז' אדר either in a regular year or if it was in a leap year on ז' אדר שני. However, this is difficult because we assume להלכה that אדר שני is אדר and אדר ראשון is the extra month. If so, Moshe did not die on the same day that he was born.
In fact, the Magen Avraham in סי' תק"פ paskens that the fast for the death of Moshe is on ז' אדר ראשון. The fact is that the Rama writes that a yahrtzeit is observed in אדר ראשון as well even though he paskens that for a bar mitzva you need to wait until אדר שני. The reason for this is either אין מעבירים על המצות of that this is what he was מקבל, and therefore the same would apply to fasting for Moshe.