Hi, @GeekBone 
I’ve found a similar and very old discussion topic (from 2016) here in the Ubuntu MATE Community - “Rename file with double click?” - regarding that possibility of renaming files on double click:
On that discussion, @ouroumov pointed out that he believes that “Caja” (“Caja is the default file manager for the MATE Desktop Environment“) does not support that feature yet (although someone could create a Python extension for Caja that would implement that) and he linked to the following issue / feature request for “Caja” (“Click on the File/Folder name to start renaming it”) in Caja’s GitHub repo / repository (issue created in the year 2014):
Reading all the comments in that issue, it seems to me that the feature is still not implemented 
In that issue, the following comment written, on 2th July 2016, by the user Etamuk, says that “Thunar” ( xfce:thunar:start [Xfce Docs] - “Thunar is a modern file manager for the Xfce Desktop Environment“) already has that double-click to rename file feature:
Click on the File/Folder name to start renaming it · Issue #333 · mate-desktop/caja · GitHub
Etamuk commented on Jul 2, 2016**
File manager Thunar has it on board. It would be nice to see this in MATE.
German readers can get more about Thunar here: [ https://wiki.ubuntuusers.de/Thunar/]
Although I have never used “Thunar”, I know that it’s definitely possible to use “Thunar” as the default File Manager in Ubuntu MATE, instead of “Caja”, and I know that a few people here in the Ubuntu MATE Community do use “Thunar” instead of “Caja” in Ubuntu MATE. See, for instance, the following discussion:
I hope this helps