No, not all 9 volt power adapters are the same. The voltage and amperage output of the adapter will vary depending on the brand and model.
Some 9 volt power adapters are regulated, which means that they will maintain a constant output voltage regardless of the input voltage. Others are unregulated, meaning the output voltage will fluctuate depending on the input voltage.
The type of adapter you need will depend on your device. If you are using devices sensitive to voltage changes, you will need a regulated adapter. If you use devices that can handle fluctuations in voltage, then an unregulated adapter will suffice.