is there a Unicode character for Copy and Paste?

How about this: &#x2398 = ⎘ which looks kind of like a copy from clipboard.

For Paste, the CLIPBOARD symbol (U+1F4CB) would be likely;
📋

My solution is to use two 📄 emojis and layer them over each other like so:

<span style="font-size: .875em; margin-right: .125em; position: relative; top: -.25em; left: -.125em">
  📄<span style="position: absolute; top: .25em; left: .25em">📄</span>
</span>

(Of course I'm a web developer, so I have access to HTML. You might need to accomplish this another way. But I'm guessing a good chunk of people looking for an answer to this UI problem are using unicode in a website.)

The neat thing about this solution is if the thing you're copying is better represented by an icon other than 📄, you might be able to switch out the emoji for something else.

The scissor ✂️ and clipboard 📋 emojis are then suitable cut/paste companions.

I use scissors character for “cut out” on site — https://unicode-table.com/

To copy I use two squares but you could also use — Two Consecutive Equals Signs or Mahjong Tile Two of Bamboos.

To paste I use - Clipboard characters. These characters correspond to characters in word.

Was looking as well, found these alternatives: ☍ ⊕ ⎘ ⩲ ⨧ ⑃ ended up using ⎘ like suggested above