A code change is required to modify how Products are found when adding them at the line level of Sales or Purchase Order, but you can change how Products are searched by default via the Catalog View or via the Popup when you use Search More.

Activate Developer Mode and search through Packagings and Barcodes so you can see the domain used:

Then customize the Search View of both the Product and Catalog Search to redefine the default search to include this tuple:

 <field name="name" position="replace">
    <field name="name" 
           string="Product" 
           filter_domain="['|', '|', '|', 
                          ('default_code', 'ilike', self), 
                          ('name', 'ilike', self), 
                          ('barcode', 'ilike', self),
                          ('uom_ids.product_uom_ids','ilike',self)]"/>
</field>

This change is applied to the Product Search when clicking Search More from a Sales Order or Purchase Order.

The same change would need to be applied to the View product.view.search.catalog to have this feature work in the Catalog View.