Copyright and trademarks are two different things. A company name might be under trademark protection, even if it is a common noun, like "Oracle". The name "Oracle" is not protected by any copyright license, and thus can not be put "under the MIT license" just by mentioning it in README.md. Nevertheless, mentioning the name in the README file might under certain circumstances violate the acceptable use according to trademark rules. Calling your database server "Free Oracle Database" in README.md is is most likely against trademark rules. On the other hand, mentioning in the same README that your product is "built on Oracle technology, but in no way endorsed by Oracle" is most likely fine, as the trademark "Oracle" is used just to refer to the company and/or product the trademark is meant to represent.
Logos and (possible depending on jurisdiction) artificially invented company names on the other hand are protectable by copyright, so a LICENSE document should make clear that the MIT license does not apply to
company_logo.png. In that case, the remaining parts of the source code may still be MIT licensed, but the complete source code repository is not, and furthermore the binary with that logo included is also not subject to the MIT license.
Combining source code you may use under the GPL with a copyrighted company logo (example: YourProprietaryVideoEditor.exe has a splash screen containing that logo, and it links against ffmpeg in a configuration that includes GPL licensed code) will result in a non-distributable binary, unless you license that logo under GPL compatible copyright terms. You still have trademark protection on that logo, though. So I could take your video editing software, fetch the logo from it and start branding fridges with it, as you most likely didn't apply for trademark protection for "household appliances", just for "software", and maybe even for "clothing" to also have merchandise protected. I also can use parts of that logo to create another logo (a work derived from your logo) and use that for my product, if the logos are clearly distinct, for example cropping a film roll from that logo to be used as a part for an entirely new logo.