Is this because of focus breathing?

A FX/Full Frame body has a 1:1 in relation to Field of View (FOV).

So a 300mm (FX or DX) lens on a FX will give 300mm FOV though you may get vignetting due to the image circle being too small for a FX sensor.

A 300mm (FX or DX) on a DX camera will have a 1:1.5x in relation to field of view. So the 300 will act like a 450mm FOV.

The issue here is the zoom lens isn't zooming like advertised .. if I read all that correctly.
 
A 200 mm lens is a 200 mm lens - regardless the camera it's mounted on.

A 200 mm lens mounted on a camera having and APS-C size image sensor will deliver the same Field Of View (FoV) as a 300 mm lens would mounted on a full frame of the same brand.

Canon APS-C image sensors are a tad smaller than Nikon APS-C sensors, which is why Canon EF-S has a crop factor of 1.6 while Nikon DX has a crop factor of 1.5.
 

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