Yes you can create your custom end point without asking for Access Token/ Authorization to get the category with its name.
Below is the steps :
create webapi.xml file in your VendorName/ModuleName/etc directory.
<?xml version="1.0"?>
<routes xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:module:Magento_Webapi:etc/webapi.xsd">
<route url="/V1/customrouter/customcategorylistingurl/" method="GET">
<service class="VendorName\ModuleName\Api\CustomCategoryListingInterface" method="getcategorylisting"/>
<resources>
<resource ref="anonymous"/>
</resources>
</route>
</routes>
Create a di.xml file in your VendorName/ModuleName/etc directory and put below code in it.
<?xml version="1.0"?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:ObjectManager/etc/config.xsd">
<preference for="VendorName\ModuleName\Api\CustomCategoryListingInterface"
type="VendorName\ModuleName\Model\CustomCategory" />
</config>
Now in your Custom Model(CustomCategory) - add the code(logic) for get all the categories with the category name then using above endpoint you will get the categories with the name.
Note : As mention above in your webapi.xml file you must pass ref as anonymous - this parameter is used for guest use which will not asked for authorization.