Can I connect multiple SN74HC595 shift registers separately (without daisy chaining them) to the same Arduino?
Yes. Each shift register typically requires three pins (data, latch, clock), but if your application makes it possible to share clocks or both clock and latch, then your required pin count will be less.

Keep in mind that if two shift registers are connected separately (no shared pins), they cannot be updated simultaneously (footnote: you can use the SPI peripheral to drive shift registers, and this can actually run while you bit bang other pins, so you sort of can update shift registers simultaneously. Sort of).

To expand on sharing latch/clock (assuming you are bit banging pins, and not using the SPI peripheral):

Shared latch: the SRs will update (the internal registers will be latched to the outputs) simultaneously. However, you will still have to clock the data to both SRs separately. The sequence would be "unlatch, clock data 1, clock data 2, latch".

Shared latch/clock: you can update the SRs simultaneously and clock the data out in a single sequence. In your clocking loop, you simply update both data pins. So the sequence is "unlatch, clock both data streams, latch". This is faster, but for many applications, this distinction won't matter.