The answers to Copy and Paste in Windows PowerShell (the link given in the other answer by Chris) suggest 5 different ways to paste text. They all work on the plain command line, but only 3 of them work for the ssh password (or when entering any text into the ssh app, such as after you've successfully logged in).
The five are:
- press right-click on the mouse
- use the mouse to open the context menu and click the Edit > Paste option
- hit Ctrl + V on the keyboard
- use the keyboard to access the context menu's Edit > Paste option: Alt + Space, then E, then P
- if the "Use Ctrl+Shift+C/V as Copy/Paste" option is activated, hitting Ctrl + Shift + V on the keyboard
The three that work for ssh are numbers 1, 2 and 5. (As the other answers mention, the ssh login prompt doesn't echo the pasted characters, and you'll need to press Enter after pasting.)
Why don't all 5 methods work?
The tool-tip for the option for number 5 gives a hint as to what's going on.

It says "regardless of input mode". The options dialog also has a new console features link, which has the following:
If you run in processed input mode, the shortcuts below apply, but in
other modes, your application must handle them
(where "shortcuts below" include Ctrl + V)
So I think that ssh runs in a mode other than "processed input mode" (and doesn't itself handle those shortcuts in the expected way) which is what apparently causes methods 3 and 4 to not work.