Return to Answer

Edit:

How to override :

Since a lot of people in comments were asking how to override.. you need to override the customer_index_index.xml in this customer_listing is referenced.

Update

Here the version for the Magento < 2.2.x :

<?xml version="1.0" encoding="UTF-8"?>
<listing xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:module:Magento_Ui:etc/ui_configuration.xsd">
    <columns name="customer_columns" class="Magento\Customer\Ui\Component\Listing\Columns">
        <argument name="data" xsi:type="array">
            <item name="config" xsi:type="array">
                <item name="childDefaults" xsi:type="array">
                    <item name="fieldAction" xsi:type="array">
                        <item name="provider" xsi:type="string">customer_listing.customer_listing.customer_columns.actions</item>
                        <item name="target" xsi:type="string">applyAction</item>
                        <item name="params" xsi:type="array">
                            <item name="0" xsi:type="string">edit</item>
                            <item name="1" xsi:type="boolean">${ $.$data.rowIndex }</item>
                        </item>
                    </item>
                </item>
            </item>
        </argument>
    </columns>
</listing>

I don't know what exactly want your client, but if the main goal is do redirect to the edit form - here the code of the customer_listing.xml file which should work in your custom module (I have tested it locally and it works fine for me):

<?xml version="1.0" encoding="UTF-8"?>
<listing xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:module:Magento_Ui:etc/ui_configuration.xsd">
    <columns name="customer_columns" class="Magento\Customer\Ui\Component\Listing\Columns">
        <settings>
            <childDefaults>
                <param name="fieldAction" xsi:type="array">
                    <item name="provider" xsi:type="string">customer_listing.customer_listing.customer_columns.actions</item>
                    <item name="target" xsi:type="string">applyAction</item>
                    <item name="params" xsi:type="array">
                        <item name="0" xsi:type="string">edit</item>
                        <item name="1" xsi:type="string">${ $.$data.rowIndex }</item>
                    </item>
                </param>
            </childDefaults>
        </settings>
    </columns>
</listing>

PS: Tested on Magento CE 2.2.1

If the main goal is redirect to the edit form - here the code of the customer_listing.xml file which should work in your custom module (I have tested it locally and it works fine for me):

<?xml version="1.0" encoding="UTF-8"?>
<listing xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:module:Magento_Ui:etc/ui_configuration.xsd">
    <columns name="customer_columns" class="Magento\Customer\Ui\Component\Listing\Columns">
        <settings>
            <childDefaults>
                <param name="fieldAction" xsi:type="array">
                    <item name="provider" xsi:type="string">customer_listing.customer_listing.customer_columns.actions</item>
                    <item name="target" xsi:type="string">applyAction</item>
                    <item name="params" xsi:type="array">
                        <item name="0" xsi:type="string">edit</item>
                        <item name="1" xsi:type="string">${ $.$data.rowIndex }</item>
                    </item>
                </param>
            </childDefaults>
        </settings>
    </columns>
</listing>

PS: Tested on Magento CE 2.2.1