I'm not aware of any tools that do this out-of-the-box. Assuming some programming experience, it would be fairly straightforward to implement this in Python using pysam (or htslib bindings for your favorite scripting language). In particular, the pileup is a very useful data structure.
Ultimately, you'll need to store an index of reference sequence positions by read name and position.
read 1, offset 0 --> chrom 3, offset 248879
read 1, offset 1 --> chrom 3, offset 248880
read 1, offset 2 --> chrom 3, offset 248882
Once you've done this for a pair of BAM files, computing the concordance statistics involves trivial set operations.
There are some important practical considerations though. If you implement the approach completely in memory, it will only be tractable for small-ish BAM files. It wouldn't take very many reads (relatively speaking) to overwhelm available RAM on a laptop or desktop. For large BAM files, you might even overwhelm the available RAM on a big-memory server. Alternatively, you could store the indexes in an on-disk database such as sqlite3. This would keep RAM requirements low, but would slow things down substantially.
I think the most reasonable approach would be to process the BAM files in batches: grab all the reads in a 100kb region (or whatever size makes sense), compute the index, store the congruence statistics, move on to the next 100kb region and repeat, and then aggregate statistics when all regions have been processed.