I know that today some people are paid every two weeks, but fortnight is a very old English word.
What would happen every two weeks (fourteen nights to be exact) in “long ago” England that made the coining of such a word useful?
Quoth the OED
Yes, but what USE is it?
“I’ll see you in a fortnight”. “I’ve been dating her for a fortnight.” “We’ve been working on this roof for a fortnight”. ETC.
How often does one use the concept of 14 days? What happens the next day when it becomes 15 days?
“I’ll see you next week.” I’ve been dating her for a month." We’ve been working on this roof for a year." These periods of time seem to be far more useful because you can “add” to them. Two weeks, three weeks; 5 months, 6 months; 8 years, 10 years, etc.
I don’t think anyone says, “I’ll see you in a couple of fortnights”.
I guess the first issue might be why ‘nights’ and not ‘days’. To wit:
No, I don’t think anyone ever says that. Personally I only really ever use it for holidays, as in “I’ve got a fortnight off in June”.
A fortnight is a half moon, i.e. the time it takes to go from full to half and half to new. That would be an important marker. Etymonline.com seems to make a reference to this under harvest:
There are many worse explanations on the Net. Sam, don’t go here unless you have a need to slam your head into a wall.
A period of fourteen nights: equivalent to French “une quinzaine” (de jours).
May be the English consider nights more important than days.
For comparison puposes, the French refer to a two week period as “fifteen days” (*J’y suis allée il y a quinze jours *– I went there two weeks ago). Even more disturbing, they refer to a one week period as “eight days” (Je te vois en huit jours – I’ll see you in a week).
Ye Gods.
I sometimes use “sen-night” for a week.
Excellent cite for the origin of “fortnight”. I shall start spreading that around. 
But only because conventionally they count today as part of the period. Like in music, a first interval is two notes with the same pitch, and a second is a one note and the next note up, &c.
Has to be the worst etymology source on the web. I can’t find a single explanation that’s correct.
An event that happens every other week can be described as occurring fortnightly. Other dialects might describe this as happening on alternate weeks, but I find that ambiguous due to overlap with ‘alternative’
It’s a reasonably useful word despite not being common in all English dialects. I hear it used quite frequently here in the UK.
I’m not here to criticize British English; I’m only curious as to when and why the term is used. If fortnight can used to describe “half a month”, then I can begin to see how it might prove to be useful. Or if it is mostly used with “about” or “almost”, I get it. However, if its meaning is strictly 14 days (like a week is strictly 7 days), then I fail to see how it could be commonly used.
But it is. “A fortnights holiday”. Government benefits are paid fortnightly. It’s just a word, it gets used when it’s appropriate to do so. I don’t see why this is a problem for you.
If it were commonly used, you’d probably see how it could be commonly used.
For an example, we use liquid measures of cups, pints, quarts, and also gallons. Do we really need all of those? Yet they’re all common. Compare week to cup, fortnight to pint, and month to quart. If “pint” weren’t common, your post could be asking the same question about that word. But it is common, so it seems normal to you (I assume).
As I noted in this post in a previous thread, many activities occur on a fortnightly basis, and the word fortnight is in constant use here. I actually used it about ten minutes ago, explaining a staff member’s absence this morning…“she has a fortnight’s leave”.
You can get a fortnightly paycheck (i.e. you get paid every two weeks).
An event can happen every two weeks. E.g. a church that practices Communion every other Sunday service could be said to have Communion every fortnight.
Like most other expressions of duration, it is sometimes used less precisely than others, but usually, a fortnight refers to two whole weeks. In my experience, it gets used to describe half a month approximately as often as a month gets approximated as four weeks.
I’m not going to go out of my way to try to defend the usage of this word - it is what it is, except that I already gave an example of how it is commonly used (that you apparently fail to see). Events that happen every other week are fortnightly.
I don’t really get the ‘what if it’s 15 days?’ thing. What happens to an hour if it’s 70 minutes, or a day if it’s 32 hours? Beyond a certain tolerance, none of these terms are useful.