In this song when the drum comes in, the accented pulse is the 4th beat out of 6 in a measure. 6/8 has the accented beats on the 1st and 4th beat, but there is no accent on the 1st beat in this song. Would its meter still be 6/8, or would it be something different like 6/4?
6/8 is probably the best option, though 6/4 would be fine considering the slow tempo. The drum hit on beat 4 is called a "backbeat" and is typical of much rock and jazz.
From the Wikipedia link, in terms of 4/4 time:
A back beat, or backbeat, is a syncopated accentuation on the "off" beat. In a simple 4 4 rhythm these are beats 2 and 4.
It’s 6/8 or less likely 6/4 for a pop song. There actually are accents on 1 and 4. The accent on 1 is implied in some measures. An accent is not necessarily simply a louder note. In this excerpt, the chord changes happen on beat one. The listener senses that right away and begins to hear that as the most important beat. The accent on beat 4 is the highest note of the arpeggio and serves as the backbeat as someone else noted.
Patterns of time signatures are just that - patterns. True, often the first beat of any bar will be signified by an accent, but not always. And - what does the signifying? Is it the bass guitar playing the root note, or the bass drum or the snare drum?
All the stresses or accents won't always fall in the same place - usuaally, in 4/4, it's the first beat that gets the accent, but in reggae, it's often non-accented. That doesn't mean we don't feel where it is - it's usually quite easy to determine still.
In 6/8, often it's sub-beats 1 and 4 that get the accent - making it more like 2/4 using triplets, which, in effect, is what it is. In this example, there's certainly an accent on beat one - no surprise there - but listen to the bass, and there's a note played on beat 6 - usually the weakest.
So, any time signature can have stresses anywhere in the bar/s, but that won't change what that time signature is called - only how it's played.