The day of the month doesn’t matter, but the month does, and you need to state this for every position that you’ve held.
Hiring managers and recruiters will look at vague dates (seasons or even just years) as an attempt to pad your resume. They might also start looking at your LinkedIn profile overall with a bit more suspicion or, even worse, close your profile all together and start looking elsewhere for candidates with more specific information.
The fact is, it may not seem like a big deal, but any time you choose to do less work you’re losing leverage against other people who took the extra steps to polish their profile. This is one of those little things that can keep you out of the discard pile.
It only takes an extra couple of minutes to make sure that you have the most specific information possible with every position.