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Override in your module /view/adminhtml/ui_component/product/product_listing.xml

<listing xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:module:Magento_Ui:etc/ui_configuration.xsd">
  <listingToolbar name="listing_top">
        <filters name="listing_filters">
                <argument name="data" xsi:type="array">
                    <item name="config" xsi:type="array">
                        <item name="templates" xsi:type="array">
                            <item name="filters" xsi:type="array">
                                <item name="select" xsi:type="array">
                                    <item name="component" xsi:type="string">Magento_Ui/js/form/element/ui-select</item>
                                    <item name="template" xsi:type="string">ui/grid/filters/elements/ui-select</item>
                                </item>
                            </item>
                        </item>
                    </item>
                </argument>
        </filters>
    </listingToolbar>
</listing>

The above code will convert all dropdowns in admin product grid to multiselect and the same can be applied to any grid in backend For example if its customer grid then use file /view/adminhtml/ui_component/customer_listing.xml and so on.