You’re not doing anything wrong; that’s how a computer works.
It’d be impossible to have a pop-up asking for your password unless the recipient has AxCrypt installed otherwise how would the computer know what to do?
AxCrypt like all other encryption software is third-party. That means you have to install its software in order to encrypt/decrypt files.
If you don’t want your accountant to download a new piece of software then you’ll have to ask him what he’s got installed on his computer (e.g. Microsoft Excel) and go out and buy that software yourself (if you don’t have it already).
Then you can share files using Microsoft’s password protection – but the result is the same – you both need the same piece of software (whether it be AxCrypt, Excel etc.) on your computer.
That’s how computers work.
Your other option is to pay for a secure cloud service (Google Drive isn’t secure) which encrypts the files and allows somebody with a link and password to download it. That assumes your accountant has a web browser installed on his computer and that you’re willing to pay the monthly secure storage fee.