Yes, that option is found in Nautilus (legacy), Caja, and Nemo.
Where is the option
The option is accessible only when using Icon View (default view) in the file manager. User will not find this option in the preferences of file manager.
In the menu bar, go to View > Arrange Items > Manually. Alternatively, right-click on mouse and select Arrange Items > Manually.
Then feel free to arrange files while using Icon View.
The arrangement persists even after changing the folder view, as long as the "Arrange items" option is not changed to something else.
Nautilus (legacy) and its original features were deprecated with the introduction of Unity and GNOME 3 desktop environment in 2010 and 2011. "Arrange Items Manually" is one of original features that were less known or used by average users.
Hence the discrepancies of Nautilus variants:
In short, the "Arrange Items" option is not available in newer releases of Nautilus (GNOME 3.6+). That leaves only Caja and Nemo, the forks of older releases of Nautilus.
Caja, the good old Nautilus
User can install Caja from the Ubuntu repositories in 16.04 and newer releases. Else use MATE for Trusty PPA by "Ubuntu MATE Developers" team to install in the older release 14.04.
Nemo, the newer old Nautilus
User can install Nemo from the Ubuntu repositories in 14.04 and newer releases. The older version depends on GNOME packages, whilst the newer version depends on Cinnamon packages.
Tested using Caja 1.12.7 and Nemo 2.8.6 in Xubuntu 16.04 (Xfce 4.12).