Does multiple cohesive Scrum Teams just should share the same Scrum Master?

Notice the word "should" instead of "must"?  There is no statement that requires the sharing of the Product Goal, Product Backlog, and Product Owner.  It is recommended.  Since there is no recommendation about the Developers or Scrum Master, I take that to imply that there is no benefit of sharing.  

I have worked where a single Scrum Team is broken up many times.  Sometimes it fits the situation described where there will be more than one team working on a single product.  But it can also happen when a single product is recognized to be more than one product and each team will then supports individual products. 

As with many things that an organization using the Scrum framework will encounter, there isn't a single solution. And what is a solution for one organization may not work for others.  In fact, that statement can apply at the individual Scrum Team levels as well.  Don't look for "the answer".  Your organization and teams should look for "the answer for this specific question based on what we know right now".  Then constantly inspect and adapt that decision as new information is learned.

Do they have have to share the Scrum Master as well or they can have more than one Scrum Master?

Have a look at the Nexus Guide and consider the possibility that a Scrum Master may be needed to facilitate product integration in addition to a Scrum Master for each team.

Each Scrum Master does not necessarily have to be a different person.

Thank you very much!