[SalesForce] SELF_REFERENCE_FROM_TRIGGER error when updating child record

To prevent a recursive call, you should make sure your trigger only executes one time. Add a class with a static boolean variable. In the trigger, have a condition that checks the value of the boolean. Once the trigger executes, change the value to false.

trigger Opportuniwith sIdentifierHandler on Opportunity (before insert,before update) {
    if(checkRecursion.runOnce()){
       If(Trigger.isInsert || Trigger.isUpdate){
                AutoNumber.OpportunityAutoNumber(Trigger.new);
       } 
    }

    }

****Utility class****

public Class checkRecursion{

        private static boolean run = true;
        public static boolean runOnce(){
          if(run){
           run=false;
           return true;
          }else{
            return run;
          }
        }
    }

After what you mentioned in the comments, it looks like updating the budget allocation causes an update to the opportunity, which is in turn in the middle of being updated by a trigger, which may cause the exception.

Perhaps you can work around this by adding future methods. Add these 2 methods to your class:

public with sharing class InsertBudgetOpp {

    // existing code still goes here...

    @future
    public static void updateChildFuture (Set <Id> oppId, Date startDate, Date endDate, Integer totalBudget) {
        Map < String, Budget_Allocation__c > baMap = updateChild (oppId, startDate, endDate, totalBudget);
        upsert baMap.values();
    }

    @future
    public static void insertChildFuture (Set <Id> oppId, Date startDate, Date endDate, Integer tb)
    {
        List <Budget_Allocation__c> baList = insertChild(Set <Id> oppId, Date startDate, Date endDate, Integer tb);
        insert baList;
    }
}

Then modify your trigger to call these future methods:

trigger EstimateBudgetPerMonth on Opportunity__c(after insert, before update) {
    Date startDate = Date.newInstance(2008, 1, 1);
    Date endDate = Date.newInstance(2008, 1, 30);
    Set < Id > oppId = new Set < Id > ();
    Integer totalBudget;
    for (Opportunity__c opp: Trigger.new) {
        startDate = opp.Campaign_Start_Date1__c;
        endDate = opp.Campaign_End_Date1__c;
        totalBudget = Integer.valueOf(opp.Budget1__c);
        oppId.add(opp.Id);
    }

    if (Trigger.isInsert) {
        System.debug('--===Insert==--#--===Insert==--#--===Insert==--');
        InsertBudgetOpp.insertChildFuture(oppId, startDate, endDate, totalBudget);
    } else if (Trigger.isUpdate) {
        System.debug('--===Update==--#--===Update==--#--===Update==--');
        InsertBudgetOpp.updateChildFuture(oppId, startDate, endDate, totalBudget);
    }
}

What this will do is, when the trigger fires, start an asynchronous method (perhaps on a different thread) that will update the child budget allocation objects. They will complete sometime in the future (generally it takes less than a minute, but Salesforce will never guarantee a timeframe). This may or may not be adequate for your solution, but it might prevent the exception.

Also note that when you end a test (with Test.stopTest()), this will guarantee that all future methods are executed.

For more on future methods, read: https://developer.salesforce.com/docs/atlas.en-us.apexcode.meta/apexcode/apex_invoking_future_methods.htm