Finding 74hc595 alternatives with the same timing- how to read datasheets?
First check the functional specifications. They are both output shift regeisters with an attached storage register/latch, so the 'broad' function is the same (but the tpic has pull outputs, versus push-pull for the hc!).

So check the functional details: type and polarity of inputs (and outputs), edges on which the action happens, etc.

I very much doubt that an Arduino (Uno at 16 MHz) could do anything fast enough to violate a timing requirement of these chips. (But a Due at 96 MHz might.)