One can mark a material as an asset in the material tab of an object. You just have to right click on the material and choose Mark as Asset. (I am using Blender 3.6, but I think this might work in Blender 3.5 too.) There is no need to open the object, then the mesh and mark the material as an asset there.
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Thanks for the tip 
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I know that this post is a little old, but I thought I’d add another way of doing it.
You can right-click on the name of the material above the node tree.
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As @zeRgenTa points out you can do it from the shader editor. Since 3.2+ you can also do this in the geometry node editor, shader editor set to world, node group node’s name field(shader, compositor, and geometry node), and object name of the object tab of the properties editor.
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Doesn’t it also have to have the file saved in an asset browser folder?
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@NP5 yes to be able to access it thru the asset browser the blend file needs to be in an assigned asset folder or a subfolder in the assigned asset folder.
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