Mainly you can achieve that with overriding the name_get method of the class/model you want to change(in this case product.product) but it seems in v13 the name_get has a nested _name_get and it is the one doing the naming.

the dirty but easy fix would be like this:

class product_name(models.Model):
    _inherit = 'product.product'

    def name_get(self):
        # TDE: this could be cleaned a bit I think

        def _name_get(d):
            name = d.get('name', '')
            code = self._context.get('display_default_code', True) and d.get('default_code', False) or False
            if code:
                name = '%s' % (name)
            return (d['id'], name)

        partner_id = self._context.get('partner_id')
        if partner_id:
            partner_ids = [partner_id, self.env['res.partner'].browse(partner_id).commercial_partner_id.id]
        else:
            partner_ids = []
        company_id = self.env.context.get('company_id')

        # all user don't have access to seller and partner
        # check access and use superuser
        self.check_access_rights("read")
        self.check_access_rule("read")

        result = []

        # Prefetch the fields used by the `name_get`, so `browse` doesn't fetch other fields
        # Use `load=False` to not call `name_get` for the `product_tmpl_id`
        self.sudo().read(['name', 'default_code', 'product_tmpl_id'], load=False)

        product_template_ids = self.sudo().mapped('product_tmpl_id').ids

        if partner_ids:
            supplier_info = self.env['product.supplierinfo'].sudo().search([
                ('product_tmpl_id', 'in', product_template_ids),
                ('name', 'in', partner_ids),
            ])
            # Prefetch the fields used by the `name_get`, so `browse` doesn't fetch other fields
            # Use `load=False` to not call `name_get` for the `product_tmpl_id` and `product_id`
            supplier_info.sudo().read(['product_tmpl_id', 'product_id', 'product_name', 'product_code'], load=False)
            supplier_info_by_template = {}
            for r in supplier_info:
                supplier_info_by_template.setdefault(r.product_tmpl_id, []).append(r)
        for product in self.sudo():
            variant = product.product_template_attribute_value_ids._get_combination_name()

            name = variant and "%s (%s)" % (product.name, variant) or product.name
            sellers = []
            if partner_ids:
                product_supplier_info = supplier_info_by_template.get(product.product_tmpl_id, [])
                sellers = [x for x in product_supplier_info if x.product_id and x.product_id == product]
                if not sellers:
                    sellers = [x for x in product_supplier_info if not x.product_id]
                # Filter out sellers based on the company. This is done afterwards for a better
                # code readability. At this point, only a few sellers should remain, so it should
                # not be a performance issue.
                if company_id:
                    sellers = [x for x in sellers if x.company_id.id in [company_id, False]]
            if sellers:
                for s in sellers:
                    seller_variant = s.product_name and (
                            variant and "%s (%s)" % (s.product_name, variant) or s.product_name
                    ) or False
                    mydict = {
                        'id': product.id,
                        'name': seller_variant or name,
                        'default_code': s.product_code or product.default_code,
                    }
                    temp = _name_get(mydict)
                    if temp not in result:
                        result.append(temp)
            else:
                mydict = {
                    'id': product.id,
                    'name': name,
                    'default_code': product.default_code,
                }
                result.append(_name_get(mydict))
        return result