No. Grammarly gets no ownership rights to your text.
By using Grammarly, you are giving us a nonexclusive license to use your content in connection with providing our services. In other words, you’re giving us formal permission to provide writing suggestions and corrections to you and to use your writing to improve our algorithms. You remain the owner of your content.
For example, if you write a novel while using Grammarly, the copyright remains under your control. That means that only you have the right to sell, publish, or distribute your novel.