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The layer is not a property of the object but rather a place it lives in. But moving an object to another layer is possible also.
Just select the objects that you want to move to another layer, select the target layer from the layer list and use "Edit->Selection->Change Layer" to move the objects to the other layer.
I guess with "layer name" you refer to the text shown in the layer list, which is basically just an alias. Currently, the shape does not know about this layer name. That is because the shape is part of the database and inside the database the layer is known by GDS layer and datatype only. The text shown in the layer list is a property of the layout view, not of the database.
It is possible however to assign layer names (usually referred to as OASIS layer names) to database layers. If you work with OASIS, those names can be stored inside the OASIS file. For GDS you need a layer mapping file. You can specify layer mapping in the reader options. With a layer mapping, the database layers can be given names as well and that name is shown in the shape properties.