I'm sorry but PM addon doesn't have such a feature, You can try to reach that by Multisite. Separating Users by multisite will let them send PM only to those Users who are in the same site. And in this case you should need to give your Users access to the sites by their membership.

I wrote briefly and with little detail, but I guess this variant can be workable.

Will it be possible for USERGROUP SILVER can only search (Profile) and message people that is in this usergroup?

Then having another USERGROUP GOLD can also do the same?

I just noticed this in the Message page where when I search for a user I see all of the members with those initials https://www.screencast.com/t/JkutSIJEAP. Members on this forum are really sensitive with their privacy and don't want to be known on each other forum.