Like below. Or we should try to make the change the original logo so it can look identical on any background?
Can the 2nd logo (inverted) harm the branding or it’s ok?
It’s easy to inverse a single color logo. But how to invert multicolored logo for dark background?
It’s not uncommon for companies to have a separate reversed/inverted logo for use on dark backgrounds. It does not go against best practices for branding to do so. As dezinezync points out, any decent branding guideline should specify how to handle it.
Rocky Mountain College’s logo exemplifies this sort of practice:
Click image for full resolution. Source: rocky.edu
Most notably, the shapes above the book are significantly changed. The bordering of the shield is also different.
If it’s a logo that you are developing, it’s perfectly fine to leave the logo unaltered for use on a dark background. The example you posted wouldn’t be considered “wrong”. However, another alternative for that logo would be to reverse the colors of the icon like so:
However, as DA01 said in a comment, if it’s not your logo then you should always check with the client to verify how to handle it.