The Shulchan Aruch (428:1) as brought in Sefaria writes,
... לְעוֹלָם בַּיּוֹם שֶׁיִּהְיֶה פּוּרִים יִהְיֶה ל''ג לָעֹמֶר, וְסִימָן פל''ג...
Purim always falls on the same day as Lag LaOmer. What is the deeper connection between the two days that they always share the same day of the week? [I would think insights would be suggested the same way others explain the connection between Tisha B'Av and Pesach falling on the same day, seen here in The Commentator's Hagada (pg 75).]
I once heard a connection as follows.
Shabbos 88b says that the Jews reaccepted the Torah on Purim.
אמר רבא אף על פי כן הדור קיבלוה בימי אחשורוש דכתיב קימו וקבלו היהודים קיימו מה שקיבלו כבר
On Lag B'Omer Rabbi Shimon allowed Rabbi Abba to write the Toras Hasod. See link here.
Thus both of these days have a strong connection to Torah.
Perhaps it is connected to the counting of seven weeks from the Korban Omer to the Shtei Halechem reflecting a gradual transcendence of physical man from a primitive, self-centered and bestial state to a state worthy of receiving the Torah. This is perhaps seen in the atypical offerings brought - barley, which is animal food, for the korban omer at the beginning of the count, and chametz, representative of physical man (including the "seor sheba'isa"-leavening midrashically identified with the yetzer hara.
The Shem MiShmuel (Lag BaOmer 5671) seems to suggest that Lag BaOmer represents the transcendent third* of the Omer count when Rabbi Akiva's students would have reached a level of mutual respect reflective of the ideal of the Torah, which is why this is when they ceased to die. Thus Lag BaOmer is the beginning of the 3rd ashmora which culminates with the giving of the Torah.
We perhaps find similar themes regarding Purim which also reflects man from within his constraints in this physical and chaotic world, transcending and acknowledging the divine. Accordingly, the Maharal explains the halacha of drinking ad d'lo yada as a way of serving G-d with one's physical self. We also find the unique mitzvah of the day of mishloach manot to create a camaraderie and to overcome the baser, selfish instincts. Finally, we see on Purim the "kiymu ma shekiblu kvar" - the willing fulfillment from within this world's constraints - of the Torah that was accepted at the culmination the ashmorah that begins on Lag BaOmer.
*paralleling the Talmudic view about the night being broken up into three ashmorot - watches.
