Can we build a new platform / cryptocurrency on top of Monero like Counterparty / Factom is built on top of Bitcoin?
The Monero blockchain can be used to send any information, as long as it's part of a transaction. There is a field (tx_extra) where you can effectively put whatever you want in there. It's really a matter of size as the limiting factor. As with the bitcoin blockchain, tx fees need to be paid (based on the size of the transaction). So, yeah, it would work in sort of the same way.