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数 refers to something's corresponding number. For example, a reading on a monitor.

量 refers to an amount, something that can be measured.

However, if you wanted to say that the number of something was increasing, (for example, the types of fruit in a supermarket) how could that be phrased? I'm guessing both terms could be used.

How does 番号 figure in here?

3

数 refers to the number of individual objects/things.
量 refers to the overall bulk amount of something.

Both can be measured.

○ バナナの数が300万本にまで増えた。
× バナナの量が300万本にまで増えた。
× バナナの数が300万トンにまで増えた。
○ バナナの量が300万トンにまで増えた。

In terms of your specific question regarding types of fruit, if you really wanted to use either 「量」 or 「数」 to refer to the number of individual types of fruit there are, it would be more appropriate to use 「数」, as you are expressing the number of individual types. However, 「種類の数が増えた」 sounds slightly unnatural/redundant, and we would simply say 「種類が増えた」.

番号 is simply a number that we humans have decided to give something for identification purposes or to express the order of something in a series.

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  • +1. I work in the engineering field here in Japan and I just wanted to add that we use 数量 very often to refer to either or both 数 and 量 when talking about amounts, numbers, volumes, etc. It makes matters a lot easier. :) – DXV Oct 25 '19 at 1:53
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    バナナって、どっちかっちゅうと「バナナの数」「X個」より「バナナの本数」「X本」って言わへん? – Chocolate Oct 25 '19 at 10:32
  • 爆笑!言いますね。途中でバナナに変えて個はすっかり見落としてもた。editします。 – sbkgs4686 Oct 25 '19 at 11:09
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To specifically answer one of your questions using your example:

...if you wanted to say that the number of something was increasing, (for example, the types of fruit in a supermarket) how could that be phrased?

It could be phrased:

スーパーにある果物の種類が増えている。 The types of fruits in a supermarket are increasing.

As @sbkgs4686 says in their answer, you don't need to add a specific word for "amount" but rather make whatever is increasing (in the example, the types) the subject of a phrase with the verb 増【ふ】える (to increase).

Aが増えている. A is increasing

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