Not about trace width
The thickness can (for manufacturing reasons) only be controlled by layer. This is typically done when ordering the pcb and is not really an issue for designing a pcb. The gerber formats used right now don’t really have the option to communicate this to your fab anyways. (It think the next generation of gerber formats will however include this information. Not sure if gerber x2 that is supported by kicad does this already.)
The only stage where you need to know the copper thickness for a particular layer during the design process is when calculating the required trace width for a high current trace. The trace width calculator of kicad therefore has a setting for the copper thickness.
Thickness also factors in RF structure design.
For most of us copper thickness is a purchase option when ordering the PCB
My usual supplier only offers 1oz and optionally 2oz/ft^2, where 1oz/ft^2 is ~35um thick
With changing copper thickness PCB manufacturer specifies different restrictions for tracks, clearances, … .
It’s also relevant for impedance control.
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