Why are counters (助数詞) necessary to describe the numerical quantity of objects in Japanese?
I'm interested in the linguistic explanation specifically, but also curious as to how strict those who are fluent tend to be about it in colloquial conversation.
Counters aren't necessary any more in Japanese than gendered nouns are in romance languages, e.g. la vache vs. le chat.
English has something extremely similar to counters called Collective Nouns. When you see several fish, you call it a "school of fish". You wouldn't say a "school of wolves", you'd say a "pack of wolves".
Are these necessary? Anyone could just say a "group of fish/wolves", but it simply wouldn't sound right to our ears. That's the same way it works in Japanese, to say 一枚のペン simply wouldn't sound right, even though you could easily say 一つのペン.
Counter words are not necessary, but many languages do use them. My favourite counter in English for illustrative purposes is "sheet(s)", as in "two sheets of paper". With other counters like "two bottles of beer", you can sometimes get away with "two beers", but "two papers" doesn't mean "two sheets of paper" (although it can now mean "two publications").
While in English counter words ("classifiers") are usually only used for mass nouns (i.e. things where it is unclear what "one [thing]" is), in Japanese counting items/people/days/etc. is always done using Japanese counter words. (One could say that Japanese has no count nouns, so that all nouns are mass nouns/non-count nouns.)
There is some level of flexibility in what counter to choose for which item. For checking which counter to use, one can consult 数え方の辞典, which gives explanatory notes to each counter, e.g.
かみ【紙】
枚【まい】、葉【よう】、本、ロール、巻き【まき】、片【ひら】、束【たば】
「葉【よう】」は葉【は】のように手に取ることができる程度の大きさで、折りたたまれていない紙を数える語です。FAX紙のようなひと巻きの紙は「本」「ロール」「巻き」で数えます。「片」はメモ用紙や紙切れ、紙吹雪などを数える語です。束ねたものは「束」で数えます。
帖【じょう】、締め【しめ】、連【れん】、丸【まる】
小売単位は「帖」を用います。半紙20枚、美濃紙【みのがみ】は48枚で「1帖」です。和紙2000枚を「ひと締め」、洋紙1000枚を「1連」といいます。「丸」は、和紙の取引単位です。
Just like we have to use a counter for paper in English. Roughly corresponding to the above, we have
a sheet of paper, a scrap of paper, a piece of paper, a roll of paper
a ream of paper, one quire of paper, one bale of paper, etc.
Adding to @bcioutier's great answer, it is somewhat wrong to assume there are counters only in Japanese or there are no counters in English as @Eiríkr Útlendi commented. I am not saying you are assuming so.
English uses a couple to indicate two. And there are a half a dozen to mean six, a dozen for twelve, a score for 20, a gross (12 dozens) for 144, etc.
There are also special words for people using prefixes for each number.
Soloist, duet, trio, quartet, quintet, sextet, septet, octet.
Your question,
Why are counters (助数詞) necessary to describe the numerical quantity of objects in Japanese?
Japanese needs counters to describe quantity of objects, some are complicated, some are not. But that's the way the language has evolved for thousands of years as English has on its own way. If someone asks why English uses duet and trio instead of just 'two people' and 'three people', you will answer because they have specific meaning to indicate the number of people in music or entertainment.
how strict those who are fluent tend to be about it in colloquial conversation.
It will depend on each native speaker. There are some counters which are strictly followed and some are not.
For more information, you can read the link on Words for numbers in English and Japanese counter word.
Why are counters necessary>> 助数詞がなかったら、「 ３歩歩いて２歩下がる」「 何名様ですか -- ３人です」とかが使えないし、「 一口ちょうだい」「 一個ちょうだい」「 一切れちょうだい」「 一杯ちょうだい」とかの違いがわからないし、「 第一球投げました！」「なにか一枚羽織りなさい」「 一発やらせろ！」とか言うときに困るからですかね・・。あと、「尾、丁、さく、切れ」とかがないと、マグロも数えられないし不便ですよね・・・。