Clustering without information about identifier
1) About clustering the products with no product information
Clustering is always possible but not always what you expected
You can cluster the products based on your dataset using the Hamming distance (if values can only be 0 or 1). Two products separated by a small distance are products that are likely to be sold in the same shops. However, there is no assurance that these products will be similar in terms of their characteristics.
Example

expensive ham (A) &amp; cheap ham (B)
expensive cheese (C) &amp; cheap cheese (D)

It is likely that shops located in wealthy neighborhoods (Shop1) will sell expensive ham and expensive cheese. It is also likely that shops in modest neighborhoods (Shop2) will sell more cheap cheese and cheap ham.
  Shop1 Shop2
A   1     0      (Exp. Ham)
B   0     1      (Cheap Ham)
C   1     0      (Exp. Cheese)
D   0     1      (Cheap Cheese)

Based on this :

A and C are very similar (distance = 0)
B and D are very similar (distance = 0)

So the clustering algorithm will (in my example) cluster products, not on their descriptions (ham vs cheese) but based on the prices. (because of shops location in this case)
I have highlighted the price-shop relation, but it is possible that some shops are clothing-only stores and thus products sold in those will likely be clothing.
Conclusion: you will discover clusters of products that are sold in the same stores but it will be difficult to understand why without further information on the shops.
2) About the (Product x Product) distance matrix clustering
The points on the resulting matrix are product duo. AB, BC, DA, etc.
The value of the points is the distance between the products of the duo. So clustering this would result in clusters of duos that have the same similarity. I don't think this makes sense.
Hope this will help.