Paralell SRAM with separate I/O ports
RAM with separate I/O certainly does exist, but is not readily available. Back 20 years ago, Cypress Semiconductor produced various RAMs with separate I/O. The closest match to your symbol was the CY7C122 and CY7C123, both 256 x 4 with separate inputs and outputs, although they also made considerably bigger parts as well. The 122 was rated at 15 - 25 nsec, while the 123 was good for 7 - 10 nsec. Power for the 122 was 330 to 495 mW, while the faster 123 was 660 to 825 mW. Here is an archived catalog from 1996 which has the parts' data sheets.

The problem with separate I/O RAM, of course, is pinout and package size. For the same size memory, an 8-bit wide RAM will require an extra 8 pins if it uses separate I/O.

As for availability, Cypress discontinued their line long ago. However, if you want to take your chances, the CY7C122 is available from China on eBay quite cheaply.