Nope.
In tibia you are unable to keep player messages and not have their name show up.
You can control names in the following ways:
Messages:
You can stop names popping up in-game, by removing on-screen messages.

Name Plates:
You can toggle through showing creature names using the CTRL+N hotkey.
It has 4 different possible states:
| Player | Everything Else |
1 | ON | ON |
2 | OFF | ON |
3 | ON | OFF |
4 | OFF | OFF |
Ignore List:
Using the ignore whitelist/blacklist you can ignore specific players and npcs too.
This has limited uses but is good for ignoring NPCS so they don't ruin clips etc.