If I remember correctly it is only with MyISAM you get instant count(*) while InnoDB, just like PostgreSQL, has to look at every row of the table (or every row of the index since InnoDB also supports index only scans). This is due to both PostgreSQL and InnoDB being MVCC (multi-version concurrency control) database engines.

And MyISAM is only fast in very specialized workloads with low write and read/write concurrency.