I just dumped my TCC database and the entries are mostly not path tied. The Chrome entry is as expected a bundle ID+designated requirement. Unless the local network permission inexplicably uses a different permissions engine, that statement is a misdiagnosis of what's happening.
$ sudo sqlite3 "/Library/Application Support/com.apple.TCC/TCC.db" "SELECT hex(csreq) FROM access WHERE client = 'com.google.Chrome';" | xxd -r -p | csreq -r /dev/stdin -t
(identifier "com.google.Chrome" or identifier "com.google.Chrome.beta" or identifier "com.google.Chrome.dev" or identifier "com.google.Chrome.canary") and anchor apple generic and certificate 1[field.1.2.840.113635.100.6.2.6] /* exists */ and certificate leaf[field.1.2.840.113635.100.6.1.13] /* exists */ and certificate leaf[subject.OU] = EQHXZ8M8AV
In other words, the permission grant applies to any Chrome channel as long as it's signed by Google's team ID.(I think both POVs are valid, but the actual solution should be for a proper OS-level interface to provide mDNS discovery...)