Isn't this implant too close to my natural tooth? (photos)

yes, its very close to next tooth .  CBCT will give perfect idea. so its my advice to get CBCT done. 

From that angle its not possible to determine the proximity of the implant to the tooth, its the wrong type of xray to determine that.  My feeling is that it should not effect the long term stability or outcome of either the implant or the natural tooth.  I think your dentist is doing a good job from what I can see.

We are looking at a panoramic x-ray which has a 25% distortion rate associatedwith it.  It is possible to see something that actually does not exist.  In my opinion I would delay restoring the implants with crowns and monitor for the next few months.  If there are changes in the appearance of the xray theni would have it removed.  It may give you a chance to keep the tooth next to it.You could also be pro active and have a CT scan done.  This could give you a clearer picture of whether the implant is in contact with the tooth.