I run a Windows 2008 server in Azure. Its public IP address is like 168.62.xxx.xxx
Is this IP address assigned exclusively to my server or is it shared among tenants like in a shared hosting.
Each Windows Azure Cloud Service gets one public IP address. Azure Virtual Machines are deployed behind a Cloud Service, so if you have several VMs deployed in the same Cloud Service your VMs will share the public IP so you'll have to keep that in mind.
Note that the IP may change over time so using a CNAME pointing to yourservice.cloudapp.net instead of pointing directly to the IP is recommended.
As @David-W stated, a VIP is dedicated to your deployment, with all virtual machines (or role instances) sharing that public IP address. One thing you may not be aware of: with Virtual Machines, you can choose between load-balanced endpoints and port-forwarded endpoints. While the former is typically used with web sites / web services (where you have, say, a web site listening on port 80 across several web server vm's), the latter allows you to directly access a specific virtual machine, differentiated by port number (maybe specific database servers in a cluster, for example).
There's a new change regarding VIP preservation and Virtual Machines: Previously, if you shut down the last Virtual Machine within a deployment (cloud service), you'd lose your VIP. As of a week ago (as announced at TechEd 2013), you can now keep a virtual machine provisioned, even when shut down, which preserves your VIP (see this post for more details).
For Cloud Services, the full rules about when a VIP changes or doesn't change is here.