Productivity Metric

Wouldn't it be better for the teams to use metrics like this to improve value delivery, and for others to gauge success by the actual value those teams produce?

My organization created new team recognition policy to identify the best team

What is the reason for wanting to identify the best team, rather than assessing each team individually?

What kind of behaviour might this encourage?

 my question is whether we can use the velocity as a metric to measure the team productivity ?

No velocity is not a measure of productivity.  Cycle time could be used though.

I am not sure how your measurement actually identifies the "best team" based on Scrum.  It seems that the better metrics would measure the value of what they provide to the stakeholders.  Because I could easily coach a team to be fast and not deliver anything of value.  

Scrum isn't supposed to make you faster.  It is supposed to help you deliver value frequently. By virtue, you will be delivering value faster but only if that is the focus and not just getting stuff done fast.  It will also uncover inefficiencies in your organization that is hindering the ability to deliver value. 

To answer you original question

Now my question is whether we can use the velocity as a metric to measure the team productivity ?

In my opinion, no.  Velocity does not indicate speed.  It indicates how much a team estimates that they can do and how consistent they are at that.  It can also be used to understand how good the team is at estimating but even then it needs to combined with something like cycle time or through put.