When copying to the clipboard in Windows, the application can choose to make several formats of the data available to an application that wishes to paste the data. I have attached a screenshot of the different formats available when copying from Chrome.

I know PySide6, a Python wrapper around Qt, lets you give the mime-type of the data you are putting into the clipboard. I don't know if there is a similar feature in pyperclip. This feature lets you share html with included hyperlinks directly with another application.
app.clipboard().mimeData().setHtml("""<html>
<body>
<!--StartFragment--><h3 class="s-post-summary--content-title" style="margin: -0.15rem 0px 0.3846rem; padding-top: 0px; padding-right: var(--su24); padding-bottom: 0px; padding-left: 0px; border: 0px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-weight: 400; font-stretch: inherit; line-height: var(--lh-md); font-family: -apple-system, BlinkMacSystemFont, "Segoe UI Adjusted", "Segoe UI", "Liberation Sans", sans-serif; font-size: var(--fs-body3); vertical-align: baseline; box-sizing: inherit; display: block; word-break: break-word !important; overflow-wrap: break-word !important; hyphens: auto !important; color: rgb(35, 38, 41); letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial;">
<a href="https://stackoverflow.com/questions/73177180/copying-hyperlinked-text-to-the-clipboard-in-python-pyperclip3" class="s-link" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: var(--theme-post-title-font-family); font-size: 17px; vertical-align: baseline; box-sizing: inherit; text-decoration: none; color: var(--theme-post-title-color-hover); cursor: pointer; user-select: auto; word-break: break-word !important; overflow-wrap: break-word !important; hyphens: auto !important;">Copying hyperlinked text to the clipboard in Python pyperclip3</a></h3><div class="s-post-summary--content-excerpt" style="margin-top: calc(var(--su2) * -1); margin-right: 0px; margin-bottom: var(--su8); margin-left: 0px; padding: 0px; border: 0px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-weight: 400; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI Adjusted", "Segoe UI", "Liberation Sans", sans-serif; font-size: 13px; vertical-align: baseline; box-sizing: inherit; color: var(--fc-medium); display: -webkit-box; -webkit-line-clamp: 2; -webkit-box-orient: vertical; overflow: hidden; word-break: break-word !important; overflow-wrap: break-word !important; hyphens: auto !important; letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial;">
My program needs to copy into the clipboard some text with hyhperlinks, so that the user</div><!--EndFragment-->
</body>
</html>
""")
Update: Pyperclip "Currently only handles plaintext." So copying hyperlinks isn't an option currently with pyperclip. https://github.com/asweigart/pyperclip#:~:text=Currently%20only%20handles%20plaintext.