This got me curious, so I looked it up, and it looks like there are surprisingly quite a few days not accounted for:
January 18 & 26
February 17 & 29 (understandably)
March 10, 11, 13, 18, & 20
April 2, 9, 13, 15, 18, & 20
May 5, 12, 16, & 21
June 4 & 6
July 12
August 31
(All of September accounted for!)
October 1, 22, 30, & 31
November 6 & 30
December 3 & 24
(Feel free to correct me about anything, as I'm only human and make mistakes.)
Personally, I share my birthday (December 29) with Vic (a cranky bull), who I'd never heard of before I looked him up lol.
And this got me curious and of course very surprised so I checked it more precisely by Months on wiki. It seems like Welcome Amiibo villagers covered some up but even if you count special characters it's still not all taken! Four days are without any birthday: March 10, 13, 18 and August 31. It sometimes happens that birthday share more than one character so these four days could be easily covered.
January 18 (Chelsea) & 26 (Vivian)
February 17 (Dobie) & 29 (Luna)
March 10, 11 (Zipper), 13, 18, & 20 (Hornsby)
April 2 (Tammi and Cashmere), 9 (Stella), 13 (Candi), 15 (Grams), 18 (Piper), & 20 (Stu)
May 5 (Paolo), 12 (Ellie), 16 (Ike), & 21 (June)
June 4 (Champ) & 6 (Lyle and Raddle)
July 12 (Kapp'n and Olive)
August 31
October 1 (Boyd), 22 (Katie and Sylvana), 30 (Wade), & 31 (Jack and Labelle)
November 6 (Boris) & 30 (Kicks and Tasha)
December 3 (Claude) & 24 (Jingle)