Yeah, anything other than hourly rate or annual salary sounds odd to me. Even hourly rates I convert to annual salaries by x2000 hours.
Maybe weekly when I’m unemployed.
But a fortnight is not the same as a half-month. 
A person paid semi-monthly gets paid 24 times in a year, while a person paid fortnightly would be paid 26 times in a year.
We don’t really have the concept of fortnight here. I thought a fortnight can be approximated to half a month. I meant “half a month” because technically that is how it is stipulated in people’s contracts. Not “two weeks” or “biweekly”. Contracts that mention other units of remuneration time are possible, but they are exceptions to the rule.
There are a few things at play right?
First, how is your pay defined? My pay from different companies has been variously defined as an annual salary, hourly rate, daily rate, annual salary plus hourly rate for overtime, and an annual salary plus a varying percentage for overtime.
Second, how are you actually paid? I’ve been paid weekly, fortnightly, and twice a month.
Third, how do you budget your money? I tend to do my initial “what can we afford?” budget plan based on annual numbers. Weekly, fortnightly, four weekly, monthly, and quarterly expenses are all converted to an annual amount and compared to annual net pay. Then I do my live budget monthly. The annual figures from the budget plan are divided by 12 and input into the monthly budget.
Finally, how do you think about your pay and pay in general? This is influenced by all of the above but could be aligned with none of them. I know my annual gross salary. I get paid fortnightly and hence I know my fortnightly net pay. I would have to calculate my monthly pay or equivalent hourly rate. I would also have to calculate my annual net pay and my fortnightly gross pay.
Some people who get paid fortnightly like to budget monthly using 2 x their fortnightly pay. If you can cover a month’s expenses with four weeks pay then twice a year you get what feels like an extra pay check. Conversely, any expenses that are paid a multiple of weeks rather than months will have an “extra” payment sometimes too.
I pay my rent fortnightly. Most months have two rent payments but some have three. If I didn’t account for this I’d get a rude shock on those bad months. So my monthly budget for rent is 2.17 times my fortnightly rent, not 2 times. It builds up each month there are only two payments so that there is enough available to cover three payments when necessary and I’m never caught short.
Me, too.
Only time I was ever a “salaried employee” was in the Navy; pay is listed at a monthly rate (and delivered twice a month).