I'm quite sure the other measures are in 3/4 time. Here's[1] a fairly typical notation (first from google image search). The previous measure would be the descending D-B-G quarter notes with the lyrics "birthday dear" (two beats for "birthday", one for "dear").
The next bar, which is notated in that image as a F-E legato slur only notates the E as a quarter note. If that was played as it is notated, the part where you say the person's name would sound very rushed. We probably started holding the E to allow time for longer names, and it just stuck.
(I find it interesting that this image doesn't even include a fermata[2] on the E; it includes the triplets on "hap-py" which is not in the original, but rushes the name.)
Also, there is a slight syncopation-like effect when she says "4" - that's the triplet effect where three notes are played in the time of two (in this case a dotted-eighth + sixteenth taking (2/3 + 1/3) of a quarter note in time). That's after the extra beat.
The next bar, which is notated in that image as a F-E legato slur only notates the E as a quarter note. If that was played as it is notated, the part where you say the person's name would sound very rushed. We probably started holding the E to allow time for longer names, and it just stuck.
(I find it interesting that this image doesn't even include a fermata[2] on the E; it includes the triplets on "hap-py" which is not in the original, but rushes the name.)
Also, there is a slight syncopation-like effect when she says "4" - that's the triplet effect where three notes are played in the time of two (in this case a dotted-eighth + sixteenth taking (2/3 + 1/3) of a quarter note in time). That's after the extra beat.
[1] http://abcnotation.com/getResource/downloads/image/happy-bir...
[2] https://en.wikipedia.org/wiki/Fermata