Yes, blister packs are recyclable (for example with machines like this one) but chances are there's no recycling company in your area that will accept and recycle it unless you are able to separate the different materials yourself.
Blister packs like the one you posted typically consist of plastic (usually PET or PVC) and aluminium. The problem with composite waste materials is that they are more difficult and costly to recycle compared to homogeneous materials, because you need to separate the different materials. Additionally plastic-aluminium composites only form a relatively small-volume waste stream, which makes it difficult to make the recycling process economical.
Recycling companies that recycle blister packs do exist, for example this US and UK-based recycler but I think that's one of the few and it seems you actually have to pay them to get it recycled (you have to buy a waste collection box and send it back once it's full).
In general it's always best to ask your local recycling company whether they recycle a particular product, and what the appropriate way of disposal is. That's because different recycling companies make different choices as to which types of waste they process and how exactly they collect and process it. If your local recycling company doesn't recycle it you could try and find another company a bit further away, but if you do find a suitable recycler keep in mind that moving trash around may cause more environmental damage than you gain from recycling.
BTW when inquiring about the recyclability of a particular material, always check the recycler's definition of 'recyclable'. Many recyclers will say they do 'thermal recycling' which means it will be incinerated.