Application Pool for Search Query and Site Settings Web Service
Application pool name: Search Service Settings.
Configurable: <domain>\svcSPESearchS

Click [OK], [OK].
Click Search Service, [Modify] (if the application hangs at “loading…”; refresh the page).
Under Databases, click Crawl Database: Search_Service_CrawlStoreDB_<GUID>, Edit Properties.
Under Edit Crawl Database change Database Name to:
<ENVIRONMENTPREFIX>_SharePoint_service_EnterpriseSearch_CrawlStore and click [OK].

Under Databases, click Property Database: Search_Service_PropertyStoreDB_<GUID>, Edit Properties.
Under Edit Property Database change Database Name to:
<ENVIRONMENTPREFIX>_SharePoint_service_EnterpriseSearch_PropertyStore and click [OK].

Click [Apply Topology Changes] (lasts >10 minutes) and Central Administration (navigationmenu top, left).
 

Rename the admin database

Backup the "old" admin database.
Create a new database with the new name. (<ENVIRONMENTPREFIX>_SharePoint_service_EnterpriseSearch_Admin)
Restore old admin database to the newly created.

Run the following PS script on the SharePoint server with a user that is db_owner on both the “old” and the new database:
$searchapp = Get-SPEnterpriseSearchServiceApplication | Where-Object {$_.name -eq "Search"}
$searchapp.Pause()
$searchapp | Set-SPEnterpriseSearchServiceApplication -DatabaseName "<ENVIRONMENTPREFIX>_SharePoint_service_EnterpriseSearch_Admin>" -DatabaseServer "<DATABASE SERVER NAME>"
Get-SPDatabase | where {$_.name -match '<OLD ADMIN DB NAME>'} | foreach {$_.Delete()};