Could not establish trust relationship for the SSL/TLS secure channel

Did you check the certificate of the Application Server as well? The search accesses the index hosted on the Application Server.

They're the same.

Some other common reasons for these types of errors are self-signed certificates or different casing between URL and certificate names. Did you check this as well?

Could have anything to do with TLS/SSL?

It's a self-signed certificate and all lowercases..

MMhh... out of ideas at the moment. It sounds like a TLS/SSL error.

 Do you have certificate pinning enabled for the app server connection inside your web portal web.config? If yes, this is precisely the error I had recently when we rotated the app server certificate without updating the pinned public key. You need to either remove (since it's deprecated with 9.1.1 anyway) or update the pinned public key in your decrypted web.config:

<add name="AppServerConnection" connectionString="QBM.AppServer.Client.ServiceClientFactory, QBM.AppServer.Client|URL=">aperture.science/.../J5EcKORTzl12345TIlO2ofB9A26cIiQ3frODo=&quot;" />

Where would I have the pinned public key? I can't find it anywhere..

You can find it in the web.config of the IdentityManager IIS application, e.g.: "C:\inetpub\wwwroot\IdentityManager\web.config" To see the connection strings unencrypted, you need to decrypt them using an admin command prompt:
C:\Windows\Microsoft.NET\Framework64\v4.0.30319\aspnet_regiis.exe -pdf connectionStrings "C:\inetpub\wwwroot\IdentityManager"
See here in this old documentation: Identity Manager 8.1.5 - Operational Guide (oneidentity.com)

Then you should have the line from my previous comment in the web.config file. You can modify it there, and run 

aspnet_regiis.exe -pef connectionStrings "C:\inetpub\wwwroot\IdentityManager"

to encrypt it again. I'd do a restart of the iis application pool afterwards for good measure.

Initially, the public key is (optionally) configured during the installation of the web designer web portal, in the step where you select the app server connection. Apologies, I can't find a screenshot for it at the moment.

How do I remove the key? Do I remove the entire "AppServerConnection" line?