Check how it is done in quotations/sale order as example

Edited

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When you put your model to inherit from the model ir.needaction_mixin, it's because you wanna indicate that there are some records that need an action and to indicate who are those records you need to override the method _needaction_domain_get to return a domain that will be used in combination with the menu action domain to filter the model records to display the count result. In the case of the sale.order it inherit from mail.thread that implements the _needaction_domain_get to display unread messages in the records.

For your case you just need to define a different domain for every menu action for the same object and of course the method _needaction_domain_get that build the domain for select the record count that need an action.

If your requirement it's not based on the domain of the menu action? that affects also your records universe in that menu, then you can go ok by specifying some different value in the context of the menus actions and use the value to build the domain in the method _needaction_domain_get based on the value of the context, something like this:

def _needaction_domain_get(self, cr, uid, context=None):
    if context.get('count_action') == 'menu1':
        return [('active_code','=','1')]
    elif context.get('count_action') == 'menu2':
        return [('active_code','=','2')]
    #else not need for menu3
    return [('active_code','=','3')]

then for the 3 menus you need to put the count_action key with the respective value.

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How to receive the action context in the _needaction_domain_get? You need to override the get_needaction_data of ir.ui.view to do that by simply copy the original code and changing the right lines, there is no other way. I highlights the changes:

class ir_ui_menu(osv.osv):
    _inherit = 'ir.ui.menu'

    def get_needaction_data(self, cr, uid, ids, context=None):
        """ Return for each menu entry of ids :
            - if it uses the needaction mechanism (needaction_enabled)
            - the needaction counter of the related action, taking into account
              the action domain
        """
        if context is None:
            context = {}
        res = {}
        menu_ids = set()
        for menu in self.browse(cr, uid, ids, context=context):
            menu_ids.add(menu.id)
            ctx = None
            if menu.action and menu.action.type in ('ir.actions.act_window', 'ir.actions.client') and menu.action.context:
                try:
                    # use magical UnquoteEvalContext to ignore undefined client-side variables such as `active_id`
                    eval_ctx = tools.UnquoteEvalContext(**context)
                    ctx = eval(menu.action.context, locals_dict=eval_ctx, nocopy=True) or None
                except Exception:
                    # if the eval still fails for some reason, we'll simply skip this menu
                    pass
            menu_ref = ctx and ctx.get('needaction_menu_ref')
            if menu_ref:
                if not isinstance(menu_ref, list):
                    menu_ref = [menu_ref]
                model_data_obj = self.pool.get('ir.model.data')
                for menu_data in menu_ref:
                    try:
                        model, id = model_data_obj.get_object_reference(cr, uid, menu_data.split('.')[0], menu_data.split('.')[1])
                        if (model == 'ir.ui.menu'):
                            menu_ids.add(id)
                    except Exception:
                        pass

        menu_ids = list(menu_ids)

        for menu in self.browse(cr, uid, menu_ids, context=context):
            res[menu.id] = {
                'needaction_enabled': False,
                'needaction_counter': False,
            }
            if menu.action and menu.action.type in ('ir.actions.act_window', 'ir.actions.client') and menu.action.res_model:
                if menu.action.res_model in self.pool:
                    obj = self.pool[menu.action.res_model]
                    if obj._needaction:
                        if menu.action.type == 'ir.actions.act_window':
                            dom = menu.action.domain and eval(menu.action.domain, {'uid': uid}) or []
                        else:
                            dom = eval(menu.action.params_store or '{}', {'uid': uid}).get('domain')
                        res[menu.id]['needaction_enabled'] = obj._needaction
                        if menu.action.context:
                            act_ctx = dict(context, **menu.action.context)
                        else:
                            act_ctx = context
                        res[menu.id]['needaction_counter'] = obj._needaction_count(cr, uid, dom, context=act_ctx)
        return res