Designing Flash Memory to USB Mass Storage bridge
You can use any type of flash memory that allows you to write at least as fast as the encoder produces data.  That one allows ~84kB/s to be written to it (3ms worst case write time, max 256 bytes per page program operation), which should easily deal with the maximum throughput of the VS1053.
Most SPI flash doesn't deal well with PC-type file systems because the minimum erase size is usually several kB and the access speed is usually not great (e.g. 84kB/s write and the sector and block erase times are horrendous).  There are minimalist embedded flash filesystems around that might do what you need, but they're so simple that it might be easier to just write your own rather than try to understand someone else's library.
There are cheap ARM MCU's with USB support and at least 2 SPI modules (1 each for the VS1053 and SPI Flash).  Given the small volume of data, USB2 Full Speed (12Mbps) is probably adequate for your purposes so something like an STM32F042 might do the job.  $3-4 in small quantities.