A reasonable consumer or customer might be confused into thinking a service named ElasticSearch(tm) Service is being provided by the company behind ElasticSearch. This confusion is exactly what trademarks rather than copyrights are meant to prevent.
The trademark law doctrine of nominative fair use allows you to describe your product as a hosted version of the ElasticSearch codebase, which provides the substance of the right you were describing, and it's also why you can describe the Toyota car you're selling as a Toyota, both without needing permission from a rights holder.
In the car case, you can also reference the product by name as a Toyota product because it is the same product Toyota sold, just being resold by you. But in the hosted service case, you're not reselling the same service as Elastic does; you're offering your own independent version of the service, backed by their technology. To prevent unwarranted damage to Elastic's reputation from any weaknesses in your service's reliability, customer support, or other factors, trademark law doesn't let you call your service ElasticSearch Service without their permission.
This works similarly for lots of software products and services, even other free and open source software projects. Debian has a trademark policy and exercises oversight of modified / derived / integrated versions shipped by the major public cloud providers to make sure that it's consistent enough with Debian's software freedom values, expected functionality, and quality standards to be called Debian, using trademark rights as the way they have that leverage.
At the same time, the cloud providers do not need trademark permission from Debian to redistribute unmodified official Debian images under the name Debian, or to derive from them without using Debian in the product name. (As with the ElasticSearch example, they can still use the word Debian in a fair and accurate way when describing the nature of any derived product they make without trademark permission.)