This is an excellent question. It is asked by the Minchas Chinuch and Rabbi Akiva Eiger and neither of them give an answer.

The problem is that there is a contradiction in the position of Rebbi Yehuda ha'Nasi (Rebbi). In our Gemara it is Rebbi who maintains that he is only allowed to cut his hair every 12 months and derives this from Batei Arei Chomah. It is also Rebbi in Erchin 31b and elsewhere who maintains that the "complete year" (Vayikra 25:30) during which the house was not redeemed, means 365 days.

(a) This contradiction is pointed out by Minchas Chinuch Mitzvah 368:12 DH v'Hineh Rebbi who asks why did Rebbi not say in our Gemara that he cuts his hair every 365 days?

(b) Teshuvas Rebbi Akiva Eiger (3:12) - cited here in Chidushei Rebbi Akiva Eiger - also points out the contradiction and adds that since one sees in the first chapter of Maseches Rosh Hashanah that Rebbi maintains that for all Halachos a year is considered as 365 days (see Tosfos Rosh Hashanah 6b DH k'Minyan) why then should the Halachah of the Nazir Olam be different from all other Halachos?

(c) Seeing as neither R. Akiva Eiger or the Minchas Chinuch give an answer to this question I suppose I could say, "So why do you expect me to have a try?" but anyway I will give it a shot.

(d) See the Shitah Mekubetzes above (end 4a in the name of the Rosh) who writes that the Halachos of Nazir Olam and Nazir Avshalom are "Hilchasa Gemiri La", i.e. these are Halachos that were passed down to Moshe Rabeinu on Har Sinai and the verses that are cited in the Gemara are merely Asmachtas. According to this one could possibly say that even though our Gemara cites the verse concerning Batei Arei Chomah, and the latter Halachah means a 365-day year according to Rebbi, nevertheless the real Limud is not from Batei Arei Choma, but rather from the Halachah l'Moshe mi'Sinai. It was said to Moshe that here 12 lunar months are sufficient, and Rebbi also received this tradition that in the particular case of Nazir one does not require 365 days. So the reason that Nazir is different from other Halachos is because Moshe Rabeinu was given a special directive to this effect.

KOL TUV

Dovid Bloom

Here is another answer to your question. Rebbi in our Gemara was not referring to lunar months but rather to solar months. So when Rebbi said in our Gemara 12 months he did in fact mean 365 days.

The fact that there is such a concept of solar months in Halachah can be seen in Rosh Hashanah 15a. There Rebbi Yanai said to Rebbi Yochanan that the New Year for an Esrog is Shevat. R. Yochanan asked him if this means Shevat of months or Shevat of the Tekufah. Rashi explains that "months" means lunar months while "Tekufah" means the sun i.e. the equinox. Rashi writes that one counts 30 days from "Tekufas Teves" (this is the winter equinox - according to Shmuel's opinion it falls on 7 January while according to Rav Ada it is on December 26) and this is the first day of Shevat of the solar system. We learn from this Gemara that there is such a thing as lunar Shevat and solar Shevat and similarly with all the Jewish months.

(This is also apparent in Rashi 11a DH she'Noldu who writes that sometimes the Nisan of the Tekufah, i.e. the lunar Nissan, enters into the lunar Iyar. See also Tosfos Rosh Hashanah 2b DH b'Chodesh. See the Sefer Sifsei Chachamim on Maseches Rosh Hashanah who cites these sources and in his introduction points out that there is such a concept in Halachah of "months of the sun." See also Midrash Rabah Bamidbar 14:18 which states that there are 12 solar months.)

See Rosh Hashanah 6b and Tosfos DH k'Minyan that Rebbi maintains that the year for every purpose is 365 days. Similarly we can now say that Rebbi takes account of solar months. So 12 months and 365 days are the same thing according to Rebbi.

KOL TUV

Dovid Bloom