Yes your concern is valid. While you do train basically the same muscles, you will be training in a slightly different way. A simple example is this: as you bend over the bar more you bring your gluts more into play then when riding upright. So in short, yes your position on the bike will change how the main muscles get worked out. Now I don't mean that you must have the exact position but get as close as you can to your position on the bike you want to perform well on.
Also, the two frames may be quite different, shorter top tube etc. saddle, etc. While it's best to use the bike that you will be riding if you make adjustments to make the hybrid as close as possible to your road bike you should be OK. I use my MTB on the trainer, it's set up for racing so I'm quite bent over but I have longer cranks etc so it's not identical but seems close enough not to cause problems transitioning to my road bike later.