[SalesForce] is there a way to copy custom objects field in salesforce

The basic problem you are facing is that your schema looks like this:

This is very hard to report against as you can't go from Program's to Tasks or vice-versa. Cross-object formula fields from Task to Program__c won't work either as you have discovered.

If you want to go with the solution of populating a picklist on Task of the Opportunity's Programs, you have a couple of choices:

Assumption - each Program__c has a Type or "Name' (but not the OOB Name field) that is what you want to report on.

Option 1 - Triggers

You will need two - one for Program__c and one for Task

The Task trigger (before insert and before update), fetches the parent Opportunity and its children Programs, setting the picklist field on Task to the list of Program__c.type__c's located

The Program trigger (after insert, after delete, after undelete), retrieves the parent Opportunity, and the Opportunity's children Tasks. The trigger then adds to the Task's picklist (if insert or undelete) or subtracts from the Task's picklist (if delete).

(A variation of this is to sync the list of Program__c's to a picklist on Opportunity)

Option 2 - Scheduled batch class

If the sync'ing isn't time crtitical, you could run a daily (hourly) batchable class that swept through all Opportunities, children Program__c, and children Tasks. Then simply smash the picklist in all of the Opportunity's Tasks to whatever the Opportunity's current set of Program__c are.

Caveats

Both solutions presume that a given Task is always associated with all Program__c for the common ancestor Opportunity.

Both solutions presume that the list of Program__c.type__c is finite and manageable via a picklist field on Task as you suggested in your OP.

@Girbot has an interesting comment and I have not tried that but populating that lookup field from Task to Program__c would still require the options above as the two lists are updated independently by user action.

When you enable field history tracking for an object then the changes are stored in a special history table for that particular object.

The naming convention for these objects is:

[standardObjectName]History e.g. AccountHistory, ContactHistory etc.

[customObjectName]__History e.g. Car__History, Brand__History etc.

The objects are not writable and therefore you won't be able to upsert them via apex. Check these fields out using workbench.