Are kanji words by themselves, or the building blocks to make words (by combining them with kana or putting two or more kanji together)?
My confusion comes from how Wanikani counts kanji and vocab differently, is that just for organisation’s sake or is it deeper than that?
Technically they can be both, but personally I think of them as building blocks to make words, just that some words are made from a single kanji 
For example, 時 can be a word on its own (とき) or used as a building block in 時間 (じかん).
Kanji have the meaning you learn in wanikani. The problem is they have different readings (Onyomi and Kunyomi). They can be used as a word if they stand alone as Kanji (Onyomi reading with exceptions), but usually they are using as a buiding block as you said, with kana or with other kanjis to create words that Japanese use as daily vocabulary!
Like 水, means water, but Japanese don’t say “すい” when talking about water alone, they say “みず”, but it’s important to learn both readings because when you learn other words like 水中 they use the kanji readings (onyomi) so they would read this as すいちゅう.
So basically the rule (with some exceptions) is, when the kanji is joint with kana, you use Kunyomi reading, if the kanji is joint with another kanji, you use the onyomi reading.
It’s a little bit a matter of perspective but I always see it as kanji being building blocks. While some words are spelled with a single kanji, not every kanji is valid as a word by itself, and the meanings ascribed to kanji are often more a theme of the words that use that kanji.
The way I like to think about it is that kanji are a way to write words. Some words you write in kana only; some with a single kanji; some with several kanji together; some with a kanji followed by kana. Some words can be written in several ways (e.g. there’s a choice of kanji). The words are what you really need to know in the end; studying individual kanji on their own is mostly useful only to the extent it makes it easier or more efficient to remember the words.
Kanji can be used as standalone words or as parts of larger words, depending on the context. WaniKani separates kanji from vocabulary to reflect this aspect of the Japanese. Kanji are characters that have meanings and various readings, while vocabulary consists of actual words used in daily conversation. This separation allows us to first focus on mastering kanji on their own, which is the central focus of WaniKani. Once we have a solid understanding of the kanji, we can then see how those characters come together to form words with the correct readings and meanings. By distinguishing between kanji and vocabulary, we as learners can effectively understand how kanji function in the language.
Sometimes, but far from most, Kanji are like letters of abbreviation, and each letters have meanings of their own. Like “asap”, each letters can be inferred to each unique words. Or “e.g.” where the reading is more predictable. (So, seeing a Kanji should remind of vocabularies for this purpose.)
More useful in the beginning to think of Kanji being just letters to form pronounceable words, and each letters aren’t reliable for exacting the pronunciation.
This begs the question: Are there kanji that never stand alone?
There are! For example, 度 is not used as a stand alone kanji, it’s always present in compound words like 速度 (そくど, sokudo) meaning “speed” or 制度 (せいど, seido) meaning “system.” Another example of this is 鬱 (うつ, utsu) — Depression, Melancholy.
Specific to here on WK when doing your reviews, the item you are being quizzed on will tell you if you are being asked for a kanji reading/meaning or a vocabulary reading/meaning, as well as changing the background colour as a cue as to what is being asked.
For multiple kanji vocabulary, or a single kanji with hiragana it will be obvious it is a vocabulary item. But for a single kanji, it could asking for the kanji or the vocab, so you need to be aware of which is being asked. Generally if you do mistake them, it will be detected and you will be given a warning (e.g. looking for the kanji reading, not the vocab reading").
Just remember that communication in Japanese is done via vocabulary (and grammar), not kanji. Kanji are part of how things are written.
You can use that standalone, where it is read たび,for example:
この度は誠にありがとうございました。
鬱 also can be used on its own (still with the うつ onyomi); one dictionary gives an example phrase
酒で鬱を散じる
I believe the two kanji in 挨拶 have no standalone uses; indeed they have practically no uses except in that particular word.
But in 酒で鬱, it’s not used as standalone, it’s describing depression derived by drinking, as for 鬱 alone is not used.
Huh? That’s multiple words, 酒 drink, で particle showing means, 鬱 depression. And 酒で鬱を散じる isn’t about “depression derived by drinking”, it’s “shake off one’s depression by drinking”. The 酒で part isn’t necessary, Kenkyusha has just 鬱を散じる “to shake off one’s depression/gloomy mood”.
鬱 is a word listed in EDICT, Progressive, Daijisen, Meikyo and Kenkyusha dictionaries.
Other examples of standalone 鬱 from dictionary examples:
留学中に鬱になったときはどうすれば？ What should I do if I become depressed while studying abroad?
彼は今鬱に陥っている he’s very depressed at the moment
鬱のときの彼女にいくら話しかけてもだめだ it’s no good talking to her when she’s feeling low
Is “A” a letter or a word ?
I feel like you’re being rather mean-spirited here. I’m aware kanji are completely different from the english alphabet, and comparing the two is foolish.
Well, technically it’s both : the letter of course, but also the determiner “a” which is, according to the Collins Dictionary, “a word which is used at the beginning of a noun group to indicate, for example, which thing you are referring to or whether you are referring to one thing or several”.
But anyways, I don’t see the point of making a comparison like this, when we learn Japanese the answer of this question about kanji is not as obvious as with the letters of the Latin alphabet.
I’m sorry it came across as blunt. I just wanted to show you an analogy between “A” and kanjis.
It’s not that “A” is a sign and a word. It’s more that it can be seen as a sign (that represent several possible sounds in English) or it can be seen as a word that contains only that letter. Depends on what you’re discussing.
Kanjis are signs that can represent both meaning and sounds, and that are used for either or both when writing words. Words that may contain a single kanji.
Just like “A”, “魚” can be seen a a sign (that usually represent the concept of fish), and a word that means fish in Japanese (pronounced さかな).
Sometimes kanjis are only used for their sound (like in 倶楽部, クラブ, a club, although in modern Japanese this writing would be rare), sometimes they are only used for their meaning (like 大人 which read おとな as a whole)
In linguistic there is the concept of “lexeme”, which is an undivisible (that is, cannot be broken in other lexemes) unit having a meaning.
A word is made from lexemes (at least one), some lexemes can stand alone as a word, but a lot (the majority?) can’t.
So, kanji are actually lexemes.
In English you have for exemple “water”, “hydr(o)”, “acqu(a)”. They are lexemes with a meaning of water. If English were to be written with kanji 水 will be used for them three (you notice there is kun reading, and two ON readings (one from Greek, the other from Latin).
And they can be used to make words, like 水防,
水疎性,
水管… That is similar of how compound words are made in Japanese (you even have the ON/kun difference)
(Note: those words are not written like that in Japanese or Chinese, but that is how they would be in English with kanji. In Japanese waterproof is 防水、but that would be read as “proofwater”… Note similar things happen in Japanese too, eg: 切腹 vs 腹切り).
The word 水 can, in English, be written with a single kanji.
That is similar to how some japanese words can be written with a single kanji.
And English also has some grammatical features that doesn’t exist in Chinese, and therefore there is no kanji for them. Plural, for example.
We could write like this:
the 水s 有 泥y
That is similar of how Japanese uses okurigana.
There are a lot.
For starting, all those kanji that only appear in a single word.
There are those words where both kanji are only used in that word, eg all those names of animals and plants that in Chinese are two syllables.
Like 蜘蛛、蝙蝠、薔薇、麒麟、檸檬
And also things like
瑠璃、珈琲、挨拶…
A single word written with two kanji with the same 部首, that is with high probability a Chinese bisylabic word for which two kanji have been created for each syllable, and are not used anywhere else (I know 珈琲/咖啡 are etymologically different, but they follow the same pattern).
For those it is better to learn the pair together. Those “paired” kanji are not lexemes, it’s the pair that is.
And also some kanji that only appear in a single word (with the other kanji that can appear in several words), like 芭 in 芭蕉
And all the kanji with very precise meanings that only appear in compounds, like 鹸、症、郵、電、員、…
It’s actually an apt comparison in that case. Just like kanji can be building blocks or standalone words, so too can “a” or “i”.