What to do if someone has the same domain as my one?

first of all, I think you can't buy the same domain name and  (And even if the domain isn't being used for anything, no, we (nor any other registrar) can't just take it for you.) All you can do is contact the owner and hope they receive the message and reply. If they don't reply, or they say no, there's not much you can do. If they are interested, then you have a chance.

Just because you have similar domain names doesn't give you any legal recourse.  To have legal recourse you need to prove infringement of some sort.

Let me provide you a prima facia example of it.... this happened a few years back.

MicroSoft sued Mr. Mike Rowe.  

Microsoft claimed infringement because of phonetic pronunciation.  Mike Rowe registered the name "MikeRoweSoft.com" which when spoken sounds exactly like MicroSoft.com.

He was sued and he had to give up the name immediately.

In your situation this doesn't apply,

On a further note:

Your site doesn't have https enabled and Google warns you not to enter the site.

Your competitor takes so long to load that the site times out.  

So fix your https and forget about the other site.

This is not logically possible as Domain name availability list is not stored with any registrar separately.

When you search for a domain, it is checking the availability on a shared/global pool. Once a registrar reserves a domain name for its customer, no other registrar/customer will be able to get it from any other source unless it is released to the pool.