How to export layer with metadata in QGIS?

There are no built-in functions to achieve what you want. Get the reference to the QgsLayerMetadata object from this answer How to get QGIS layer information saved in JSON format? and as stated in comment, write your own metadata as JSON in a separated file.

Alternatively, you can write the layer definition with something like below. It will contains several information including metadata in an XML file

layer = iface.activeLayer()
# mylayer.qlr is hard coded here but it could be determined from "layer.dataProvider().dataSourceUri()"
# if it's a filename or the layer name "layer.name()" or layer id "layer.id()"
with open('mylayer.qlr', 'w') as f:
    f.write(QgsLayerDefinition.exportLayerDefinitionLayers([layer], QgsReadWriteContext()).toString())