You certainly can have shared private/public keys/address between multiple blockchains. First you should ensure when creating the second (or subsequent) chain that the following blockchain parameters match those in the first chain: address-pubkeyhash-version, address-scripthash-version, private-key-version and address-checksum-value. You need to set these of course before starting the chain.
Then you can use the dumpprivkey and importprivkey commands to extract private keys from a node on one chain and then import them into a node on another chain. You will then have the same address usable on multiple chains, and it will also have the same private key.