I believe this is caused here: https://github.com/PowerShell/PSReadLine/blob/v2.1.0/PSReadLine/Clipboard.cs#L72-L82

On my WSL Ubuntu, the following returns true:
[System.Runtime.InteropServices.RuntimeInformation]::IsOSPlatform([System.Runtime.InteropServices.OSPlatform]::Linux)

I have no xclip binary that I can find with which xclip.

clip.exe does work.

I wonder if this is sufficient to disambiguate?
[System.Runtime.InteropServices.RuntimeInformation]::OSDescription
Linux 4.19.128-microsoft-standard #1 SMP Tue Jun 23 12:58:10 UTC 2020

I offer this as a workaround:

Set-PSReadLineKeyHandler -Chord Ctrl+c -ScriptBlock {
    [int]$SelStart = 0
    [int]$SelLength = 0
    [Microsoft.PowerShell.PSConsoleReadLine]::GetSelectionState([ref]$SelStart, [ref]$SelLength)

    if ($SelStart -eq -1 -and $SelLength -eq -1)
    {
        return [Microsoft.PowerShell.PSConsoleReadLine]::CancelLine()
    }

    [string]$Buffer = ''
    [int]$Cursor = 0
    [Microsoft.PowerShell.PSConsoleReadLine]::GetBufferState([ref]$Buffer, [ref]$Cursor)

    $Buffer.SubString($SelStart, $SelLength) | clip.exe
}

Set-PSReadLineKeyHandler -Chord Ctrl+x -ScriptBlock {
    [int]$SelStart = 0
    [int]$SelLength = 0
    [Microsoft.PowerShell.PSConsoleReadLine]::GetSelectionState([ref]$SelStart, [ref]$SelLength)

    if ($SelStart -eq -1 -and $SelLength -eq -1)
    {
        return
    }

    [string]$Buffer = ''
    [int]$Cursor = 0
    [Microsoft.PowerShell.PSConsoleReadLine]::GetBufferState([ref]$Buffer, [ref]$Cursor)

    $Buffer.SubString($SelStart, $SelLength) | clip.exe

    [Microsoft.PowerShell.PSConsoleReadLine]::Cut()
}

I don't have a way to avoid the trailing newline chars.

This may be a better workaround.

Create /usr/local/bin/xclip with the following contents (and grant sudo chmod a+x /usr/local/bin/xclip):

#!/bin/sh

# https://github.com/PowerShell/PSReadLine/issues/1993
# this is not an xclip replacement! It does just enough to let PSReadline work on WSL.

if [ "$3" = "-out" ]; then
    # paste
    if command -v pwsh.exe >/dev/null 2>&1; then
        pwsh="pwsh.exe"
    elif command -v powershell.exe >/dev/null 2>&1; then
        pwsh="powershell.exe"
    else
        exit 1
    fi
    pwsh="$pwsh -NoProfile -NoLogo -NonInteractive"

    $pwsh -command Get-Clipboard | sed 's/\r$//' | head -c -1
    exit 0

else
    # cut or copy
    tee <&0 | clip.exe
    exit 0
fi

For reasons I don't understand, it doesn't suffer from the annoying trailing newline that clip.exe normally appends.