UNIX time uses a Julian calendar date as a reference, but is independent after that.
As for the 13 month calendar, it’s about as nice as cloverleaf interchanges: appealing because it’s symmetrical, terrible in practice. Having the days of the month always align to the same weekday means leap years would make things even worse because every 4 years the entire calendar shifts. And if you skip the leap day as a holiday then you just make calculating dates from an epoch like UNIX time even more convoluted.