First with domain names. The domain "nissan.com" is not owned by the well-known car company but by a completely unrelated computer company. As "Nissan Motors v. Nissan Computer" settled, this is totally fine and Nissan Computer still owns the domain.
Besides exact matches there are also similar-looking names. For example, a student named Mike Rowe started a small webdesign company called MikeRoweSoft, which drew the attention of Microsoft, leading to "Microsoft v. MikeRoweSoft" - which was settled out of court and resulted in the domain being transferred to Microsoft.
Second are Extended Validation domains - which used to show the company name in the URL bar. As Ian Carroll demonstrated[0] this isn't really worth a lot, and browsers no longer bother showing it at all[1].
Company names also often overlap when they are active in different areas, such as Apple Corp (record label founded by The Beatles) and Apple Inc. (tech multinational) - which over the years have shifted towards a rather impressive market overlap! Some companies are split with both sides keeping the original name, such as Motorola Inc.'s split into Motorola Solutions and Motorola Mobility. Sometimes products are sold under a completely different brand name, such as HMD selling Nokia-branded smartphones, or TP Vision selling Philips-branded televisions while MMD sells Philips-branded gaming monitors!
The thing is, reality is just too complicated for a "very simple" register. How are you supposed to fit in all of the scenarios listed above while still keeping it usable?
[0]: https://arstechnica.com/information-technology/2017/12/nope-... [1]: https://www.troyhunt.com/extended-validation-certificates-ar...