If you want for update operation also simply add  before update  like

trigger TriggerExistingRelation on SFDC_Contact_Connector__c (before insert,before update) {

}

But as per your requirement entire code is wrong. It won't work for Insert oparetion also.

Why because your requirement is to check the  two records are same or not, But here in if condition your writing code for taking all record values but not perticular record.

 //Get the list
    List<SFDC_Contact_Connector__c> relrecs = [Select cl.Contact__c, cl.Related_Contact__c from  SFDC_Contact_Connector__c cl where cl.Id  IN :relids];
    
    set<id> ContactsSet = new set<id>();
    for(SFDC_Contact_Connector__c cl: relrecs)
    {
        ContactsSet.add(cl.Contact__c);
    }
    set<id> RelContsSet = new set<id>();
    for(SFDC_Contact_Connector__c cl: relrecs)
    {
        RelcontsSet.add(cl.Related_Contact__c);
    }

 if(ContactsSet.contains(cl.Contact__c) && RelContsSet.contains(cl.Related_Contact__c))

for Example:-

cricket is junction object

Batsman and Bowlers are parents

Here i am creating two records in Cricket object

Cricketname1

sachin

Zaheer

Cricketname2

dravid

sreenth

The Next record should be allow if it is like this

Cricketname3

sachin

sreenth

But your code Doesn't allow this record why because it is checking all two record values

So here you have to check the values are in the same record are not

At the time only   Cricketname3  record will allow.

So instead of set use map here

Map<id,id> ContactsSet = new Map<id,id>();

Map<id,id> RelContsSet = new Map<id,id>();
    for(SFDC_Contact_Connector__c cl: relrecs)       // use one loop is enough
    {
        ContactsSet.put(cl.Contact__c,cl.id);

        RelcontsSet.add(cl.Related_Contact__c,cl.id );
    }

So here Key is Different But Values are same

In if condition based on value(record id) you can check and then only return error

Better to use System.Debug();       For every step then your can find the errors easily

See some Map examples and follow up

Prem