As にち, 日 doesn't seem to be a counter (which is always preceded with dash in dictionaries). Does this mean that 日 means Sunday when used with days of the month? If so, how does it make sense?
Is 日 a counter in 十三日 or is it not?
So, I've re-read your question and the following answer by naruto, and I think I see the disconnect in your understanding.
When speaking about Kanji in general, you need to think of it in two separate ways. One is how you read it. The other is its meaning. We'll start at reading Kanji, then I'll explain deriving a meaning from context.
All Kanji can have one or more readings. These readings are grouped into two categories: 音読｛おんよ｝み and 訓読｛くんよ｝み. The short form of this explanation is that 音読み are readings for a Kanji that are derived from Chinese. Usually these readings will show up in compound words utilizing various Kanji, but this is not an exclusive rule, just a general one. By contrast, 訓読み are readings for Kanji native to Japan.
In the case of the Kanji 日 there are actually many readings:
音読み (in Katakana to emphasize Chinese origin): ニチ、ジツ
Example words using these readings:
ニチ： 毎日｛まいにち｝： Every day
ジツ： 本日｛ほんじつ｝： Today
訓読み (in Hiragana to emphasize Japanese origin): ひ、-び、か
Example words using these readings:
ひ： 日｛ひ｝： Day
-び： 記念日｛きねんび｝： Anniversary
か： 十日｛とおか｝： Ten days, or, the tenth day of the month
In addition to this, there are a few exceptions and special words, which usually are native Japanese words, that are approximated in writing using Kanji, despite their reading. For example:
As you can see, this word contains the 日 Kanji but is not read in any of the readings covered within the 音読み and 訓読み. You'll come across a few of these.
Be aware that the 日 Kanji has several meanings all at once. The most common are:
day, sun, Sunday, (counter for days)
When attaining fluency in written Japanese, often you'll encounter a mix of Kana and Kanji in writing. Because Kanji can have so many readings and individual meanings, it's important to be aware of vocabulary, and also context in use. This is where I think the issue is here for you.
Consider these example sentences for the Kanji 日:
Today is December 7th.
The 25th day of this month is Christmas.
Now consider these other examples:
There are seven days in a week's time.
If you drove from California to New York, it would take about two days' time.
As you can see, though in the above examples the word 七日｛なのか｝ appears written the same way, read the same way, but has two different meanings, it's clear to see which meaning is intended by context. Things that clue me in are verb usage, as well as the presence of a date. These are the kind of context clues you need to be aware of.
The overall simplified meaning for the Kanji 日 is actually Sun. Many of the meanings for 日 that you'll find in a dictionary are derived from the fact that 日 means Sun.
日 as "Day": The daytime is when the sun is up. Therefore, a day is counted by the cycle of the sun rising and setting.
日 as "Sunday": This Kanji literally only means Sun. However...
The complete word for Sunday is actually 日曜日｛にちようび｝. It just so happens that when writing a date in Japan, often it is given in the format of YYYY年MM月DD日 (day of the week). For example, today is December 7th, 2017, and it is Thursday. You'd see this as: 2017年12月7日（木）.
So, the 木 inside parentheses is short for 木曜日｛もくようび｝, the same way that 日 is short for 日曜日.
If you mean something like "12月24日", it just means "December 24th". 日 is simply a counter-like suffix that follows a day of a month, just as 月 is the suffix for months and 年 is the suffix for years.
日 also means Sunday, but it does not directly follow a number. If you see something like "12月10日 (日)", the second 日 in parentheses means "Sunday".