Think of biweekly and biweekly as homonyms, they can mean either and you figure out the meaning through context.
Very few things happen twice a week, biweekly usually means every second week, but it's never used because fortnightly is preferred.
Others here are saying bimonthly means twice a month but I've never heard it used that way. Again, very few things happen twice a month, it's always fortnightly which is not the same. Lots of things happen every second month, "the board meets bimonthly", that means 6 times a year.
Biannual always means twice a year because what things do you do every second year?
In all cases you can use the alternative meaning like "I visit my cousin biannually" and it's not incorrect but of course "I visit my cousin every second year" avoids confusion.
I'm a tax consultant. I look at what companies pay people all day every day. I've never seen a company pay twice monthly. Always fortnightly. This might vary by region but unheard of here.
The word for an occurrence of every two years or for a duration of two years is biennial. Plenty of events are biennial, such as festivals, exhibitions and conferences. The Olympics and Football World Cup are quadrennial.
Just to note, while fortnightly is used frequently in many countries, it is almost never used in the US, which I think is what contributes to the posters confusion (assuming they are from the US) .