This is a SP2013 issue:
I've placed a basic list view webpart onto a page. I try enabling the "view selector menu" to show from the webpart properties. When I apply the changes, the view selector shows while still in edit mode. When I save the view, the option reverts back to disabling the view selector. I can figure out why for the life of me. I have tried with both SP2013 preview and an on-prem and get the same exact issue (both OOB masterpages and a custom one). Any ideas?
My SharePoint 2013 is RTM version(15.0.4420.1017) which could enable the list view web part property "View selector menu", please check and make sure your SharePoint version is the same.
TechNet Community Support
Yes, you are right. that is what i mentioned earlier post, it seems like a bug. However, if you use Pages Library as in List View web part, View Selector is working fine but for other list and libraries it wont.
I think you need to extend this OOTB List view web part and through coding override the "Disable View Selector menu" property to make it work.