For anyone stumbling on this question after years, I tried Nyerguds' answer above, and while it did work for many programs such as Discord, it did not work out for pasting transparent pictures in Adobe Photoshop 2024 (25.12.3).

However, I did notice Photoshop will successfully paste transparent pictures copied from Google Chrome (but not Firefox).

For this reason, I have tried to investigate this by debugging the clipboard state, and while I don't have Nyerguds' intrinsic knowledge to really understand about what's going on internally in terms of format, here's a version that should work as a starting point for pasting transparent pictures in Photoshop.

You need to add a "Format17" entry to the DataObject (also known as DIBV5), and the data must use the BITMAPV5HEADER with the bV5AlphaMask value set to to 0xFF000000 for Photoshop to acknowledge it as being a transparent picture.

The code below assumes you are already using Nyerguds' answer.

// byte[] dibData is the first 13 lines of the `public static Byte[] ConvertToDib(Image image)` code from Nyerguds's answer.

private static byte[] ToDibV5(byte[] dibData, int imageWidth, int imageHeight)
{
    const uint dataStartPos = 0x7C;
    var result = new byte[dataStartPos + dibData.Length];
    
    // Lines with IGNORED means the value of 0 is used.
    
    // https://learn.microsoft.com/en-us/windows/win32/api/wingdi/ns-wingdi-bitmapv5header
    // 4 DWORD biSize;
    ArrayUtils.WriteIntToByteArray(result, 0x00, 4, true, dataStartPos);
    // 4 LONG  biWidth;
    ArrayUtils.WriteIntToByteArray(result, 0x04, 4, true, (uint)imageWidth);
    // 4 LONG  biHeight;
    ArrayUtils.WriteIntToByteArray(result, 0x08, 4, true, (uint)imageHeight);
    // 2 WORD  biPlanes;
    ArrayUtils.WriteIntToByteArray(result, 0x0C, 2, true, 0x01);
    // 2 WORD  biBitCount;
    ArrayUtils.WriteIntToByteArray(result, 0x0E, 2, true, 0x20);
    // IGNORED -- 4 DWORD biCompression;
    // IGNORED -- 4 DWORD biSizeImage;
    // IGNORED -- 4 LONG  biXPelsPerMeter;
    // IGNORED -- 4 LONG  biYPelsPerMeter;
    // IGNORED -- 4 DWORD biClrUsed;
    // IGNORED -- 4 DWORD biClrImportant;
    // IGNORED -- 4 DWORD        bV5Size;
    // IGNORED -- 4 LONG         bV5Width;
    // IGNORED -- 4 LONG         bV5Height;
    // IGNORED -- 2 WORD         bV5Planes;
    // IGNORED -- 2 WORD         bV5BitCount;
    // IGNORED -- 4 DWORD        bV5Compression;
    // IGNORED -- 4 DWORD        bV5SizeImage;
    // IGNORED -- 4 LONG         bV5XPelsPerMeter;
    // IGNORED -- 4 LONG         bV5YPelsPerMeter;
    // IGNORED -- 4 DWORD        bV5ClrUsed;
    // IGNORED -- 4 DWORD        bV5ClrImportant;
    // IGNORED -- 4 DWORD        bV5RedMask;
    // IGNORED -- 4 DWORD        bV5GreenMask;
    // IGNORED -- 4 DWORD        bV5BlueMask;
    // 4 DWORD        bV5AlphaMask;
    ArrayUtils.WriteIntToByteArray(result, 0x34, 4, true, 0xFF000000); // MUST BE SET for alpha to work in Photoshop.
    // IGNORED - 4 DWORD        bV5CSType;
    // ArrayUtils.WriteIntToByteArray(fullImage, 0x38, 4, true, 0x57696E20); // Google Chrome sets this to LCS_WINDOWS_COLOR_SPACE(0x57696E20)
    // CIEXYZTRIPLE bV5Endpoints;
    // IGNORED - 4 DWORD        bV5GammaRed;
    // IGNORED - 4 DWORD        bV5GammaGreen;
    // IGNORED - 4 DWORD        bV5GammaBlue;
    // IGNORED - 4 DWORD        bV5Intent;
    // ArrayUtils.WriteIntToByteArray(fullImage, 0x6C, 4, true, 0x04); // Google Chrome sets this to LCS_GM_IMAGES(0x04)
    // IGNORED - 4 DWORD        bV5ProfileData;
    // IGNORED - 4 DWORD        bV5ProfileSize;
    // IGNORED - 4 DWORD        bV5Reserved;
    
    Array.Copy(dibData, 0, result, dataStartPos, dibData.Length);
    
    return result;
}

// Then, add this in your existing DataObject.
data.SetData("Format17", false, dibv5Data);

// If by any chance you are using SetClipboardData in user32.dll, use `(uint)17` instead of `"Format17"`.