Indeed, noise is variable at any given ISO, influenced by many different shot conditions. First off, exposure significantly affects noise - the better exposed, the less noise - the less shadow area in a shot, the less noise - the more light overall, the less noise. You will find noise most prevalent in shadow areas and darker colors than lighter ones - an overexposed blown out shot will have virtually no noticeable noise. Nailing the exposure is very important to control noise.
Also, color makes a difference, as different sensors can have worse noise in a particular color range. For example, many cameras struggle with green channel noise, some have it bad in the blue channel, some are very poor in reds. How the camera's internal noise reduction settings are set if using JPG will affect the noise vs. detail that you'll get, often with a particular color range a bit more 'smeared' due to heavy noise reduction trying to clean it up.
Even air temperature plays a part - sensors become noisier as they get hotter - so taking photos at higher ISOs after heavy shooting, or taking photos in hot climates, will result in more noise than a cold camera turning on for the first time or shooting in cool weather.