You are not pasting plain text, but text with multi-markdown styling syntax in it. Pages is not designed to detect and convert multi-markdown styling hints. But all is not lost…
Copy the multi-markdown content to the clipboard, and run the following Shortcuts workflow to convert the multi-markdown content into Rich text. Then paste that Rich Text into Pages and you will have bold and italic styling.

and after pasting into Pages v14.3 on Sequoia v15.3.2:

when the following was copied to the clipboard beforehand:

The bold and italicized words could also be surrounded by double underscores.