I understand in SharePoint 2013 there is a Project server.  The Project site is an expansion from MS Projects which allows multi-user co-authoring and collaboration and a site to store documents.   We will need project management but is the Project Server required for these enhancements? or is it just for scalability purpose it's a good idea to have a project server?

Understanding of these requirements will be useful for hardware planning.

Project Server is just required when you want to use Microsoft Project. 

It's definitely not a requirement for SharePoint.

Hi 

No but it is other way around. As Project Server 2013 is built on SharePoint Server 2013 so while installing Project server 2013 you need SharePoint 2013.

But You do not need Project Server for installing SharePoint

It's other way round. Project server 2013 requires SharePoint Server 2013 as a pre-requisite.

If you want to manage project at individual project levels, you can use project site feature. It will provide you a nice timeline, gantt chart along with some other features in a SP site.

However, Project Server provides centralized management of all projects in your enterprise. You can have centralized management of resources, forecasting, centralized calendars, various project templates to start with, OOB reports like burndown/resource overview which your PMO team would be delighted to use.

https://products.office.com/en-us/project/project-and-portfolio-management-software

Hope this helps.