day vs day of the month

day and day of the month both are nouns.

Word	Noun	Adjective	Verb	Adverb
day	Yes	No	No	No
day of the month	Yes	No	No	No

As nouns, day of the month is a hyponym of day; that is, day of the month is a word with a more specific, narrower meaning than day:

Other hyponyms of day include tomorrow, today, yesterday, morrow, eve, date, date, snow day.

day (noun)	day of the month (noun)
a period of opportunity	the specified day of the month
time for Earth to make a complete rotation on its axis
a day assigned to a particular purpose or observance
the time after sunrise and before sunset while it is light outside
a point or period in time
the recurring hours when you are not sleeping (especially those when you are working)
an era of existence or influence
the time for one complete rotation of the earth relative to a particular star, about 4 minutes shorter than a mean solar day
the period of time taken by a particular planet (e.g. Mars) to make a complete rotation on its axis