Not at all - old wiring had only two wires, the power and neutral, but modern wiring has three, power, neutral and ground. Deleting the ground makes your modern equipment prone to failure during a power surge and increases the likelihood of a fire. The little wire loop at the bottom of the adapter is supposed to ground to the screw in the middle of the receptacle cover, but that only works if it has never been painted over, has no corrosion and you use it in the top outlet and even then you must trust that there is no dirt or corrosion where the screw goes through the receptacle behind the cover or you might not get good contact and make sure it touches the screw and maintains solid contact at all times. In a perfect world, it should work. In the real world, it hardly ever does.
If you have old wiring, you probably have metal electrical boxes, which means that you can pretty easily upgrade your receptacles without running new wire by grounding them to the boxes (modern boxes are plastic and you cannot ground to them). Not saying leaving old wire is the greatest thing in the world, and of course a grounding spike is a much better ground than the box, but a ground is always better than no ground and if you can't afford to update the whole house, updating the receptacles is better than nothing. Remember that any damp/wet application requires a GFCI style receptacle, that means kitchen, bathroom, washroom if you have one, outside and though it is not required, I use them in my basement because we get water in there sometimes.
Please remember that working with electricity is dangerous. If you don't know what you're doing, call a professional.