I was wondering about if it’s possible to use different thickness’ of copper on a pcb as a measure to refrain from using resistors.
It is not practical using conventional PCB technology
The dimensions required to get values you would typically want would be impractical- the resistivity of copper is too low. This is not accidental, it's a good part of why we use copper as a conductor material.
Even if you could get a value you needed, the tolerance would be terrible, and trimming equipment is not cheap.
Copper is a poor choice for a resistor element- not only is the resistivity too low, it changes resistivity greatly with temperature (about 3900ppm/K) so it's more like a temperature sensor than a stable resistor.
On top of that it tends to corrode in the presence of some typical pollutants so thin narrow structures would change in resistance.
It is possible in some cases to print thick film resistors on PCB substrate, but the PCB material limits the curing temperature so the resistors won't be very good (stable), and the tolerance won't be good either. Even the very cheapest SMT resistor is fired on a ceramic substrate at high temperature, trimmed to within a tolerance and coated to protect the element.
Sometimes a very low value appears tempting to make with a PCB trace such as for a crude current limit shunt. Usually it's best to 'resist'.
Stop and actually think about it.
First, look up the minimum trace width your board house supports. Now figure the resistance of that per mm or whatever. You will quickly see there is no way to get anything more than "low" value resistors this way.
Go figure out the length required just to achieve 1 kΩ. No, actually do it. Then figure the smallest area that length trace can be packed into, using the smallest line and space width your board house supports. Now compare the cost of that board area to the cost of a common 1 kΩ 0402, 0603, or 0805 resistor.
Now consider how you'd go about getting higher values. 100 kΩ would require 100 times the board area, and resistors higher than that are used routinely.
Another problem with copper trace resistors is their temperature coefficient, even if you need a value low enough to be reasonably realizable with a trace.
Then your question is really asking about thickness, not width as what I answered for above. You don't get to change the width of a plane in different places. Unless you use some very unusual (and therefore expensive) process, each plane of a PCB is a fixed width. The common default is "1 ounce" copper, meaning one ounce of copper per square foot. You an do the math and convert that to a thickness equivalent.