Is it possible to exist many users with the same name? (In a particular case, one registered 1 year ago, and other recently.)
Yes. Usernames are not distinct or required to be unique. Registration information must be unique, but the actual display name has no mechanical restrictions. Thus, if someone just wants to be known as "John", they can do so no matter how many people have already done that.
Here, a lot of users like to use their full names or otherwise personally identifiable callsigns. So this probably doesn't come up as often here. It is still possible for users to inadvertently happen to share names, though.
We do request name changes if a user attempts to share a name with a Stack Exchange employee such as myself or a mechanical process like Community. For this reason, Stack Exchange employees will generally attempt to make sure their names are unique so that fewer people run into this accidentally. Impersonating a moderator of a site is also generally a bad idea, and likewise we recommend that moderators make sure they have identifiably unique names when possible, at least within their own site.
There is nothing preventing different accounts from having the same name. You can check for pre-existing nicknames by searching on the Users page (using the "Type to find users:" box).
For example, there are at least 13 registered accounts with the name "Rob" at this time. I don't know if that list contains unregistered accounts.