Does Adobe Illustrator have a feature to warp a group of shapes similar to Photoshop's
So far I've tried the reshape tool seems to only work on individual shapes?
Object > Envelope distort > Make with mesh is almost certainly what you want.
The 'Envelope distort' idea in its many flavours is all about placing an object over the group, then, transformations applied to that object apply to the group (including differences in shape compared to a plain old square bounding box). Meshes are particularly good envelopes as there's no initial distortion but lots of fine control over the distortion you apply - but any object can be used as an envelope. It's really powerful and worth taking some time to experiment with.
Lots of good tutorials out there, here's a good one to get started with, and another that does a good job of listing the various flavours of envelope distort (some images previewed below):