This is a bit like asking "how do I give two files the exact same name, and put them in the exact same folder on my computer?" Just like files, separate meshes must have unique names... otherwise, neither you nor Blender knows which mesh is which.
Firstly, lets define our terms. A "Mesh" is a collection of vertices. Meshes have to live inside an "Object." An "Object" is a container for vertices. An object can contain literally any arrangement of vertices; connected, unconnected, polygons, individual points, etc. All those vertices are considered one mesh when they are inside an object, even if they aren't connected to each other. You can think of vertices a little like letters. If vertices -> letters, then polygons -> words, text -> mesh, and object -> document. (It doesn't work perfectly because you don't name your text separately from the document it's in, but Blender does do this with meshes and objects).
Each object must have a unique name, and each mesh must have a unique name. Typically, a mesh gets its first name from the object it's created in, but it doesn't have to.
To answer your question, you have two options. You can use ctrlj to join the meshes in any selected objects into one object (Blender will put them in the "active" object, and then delete the empty ones). This takes all the separate collections of vertices and puts them in the same container. This also gives them all the same mesh name... because they are all one mesh. To continue our files analogy, this method would be like cutting the text from one document and pasting at the end of another other document, and deleting the empty document. You now only have one container, but you still have the text from both documents.
Your other option is to use some other method to give unique names. Maybe it makes sense for your meshes to be in separate objects. In that case, there's likely a descriptive way to name them that will also uniquely identify them.