Why am I cleared to use the elliptical but not the stationary bike?

You should be fine to use bike unless the concern is your Blood Pressure being too high.  Avoiding arm strenuous activity is the way to go and I agree with you thet avoiding the elliptical would be the right thing to do.  Discuss with your PS.

Best to clarify directly with your plastic surgeon who knows exactly what procedure was performed and is ultimately responsible for your care. Best wishes.

Seems like there is more to your question but a stationary bike (not those cycling classes) should be no worse than an elliptical.  You should ask your surgeon for some clarification.  In my practice, patients can start using such devices 1-2 weeks post-op, starting slowly then building up as tolerated.

With the explosion of SoulCycle and other spin classes available and popular right now, I'm not sure that I would ok a patient of mine to take one of these rigorous classes just 2 weeks after breast augmentation. These classes have a lot of moving mechanical parts, require a lot of movement on the bike, including a lot of push ups and clients usually sweat profusely. None of that is good for some just 2 weeks out from a breast augmentation. However, if you are talking about sitting on a recumbent stationary bike at your own leisure, I certainly see your point. Best to call your surgeon and clarify this with him or her directly. 

It is uncertain as to why you aren't released to ride a stationary bike since it is less impact than the elliptical. But because you are questioning the restrictions it is best to contact your PS office to find out the reasons for that particular restriction.