Most of the power dissipation will be from the Vdd current in your application. If you run it at 12V Vdd, say, note that it can draw as much as 15mA, which is 180mW while clocking it at 8MHz, according to the datasheet. It's not clear (to me anyway) what the Vdd current draw is otherwise or whether there is a preferable state for the input lines. The datasheet is not very helpful, I'm afraid. For example it refers to Vin = 0, but there is no pin called Vin. It's possible the dissipation under static conditions is worse than 180mW but I would guess not (and verify!).
Edit: I looked at a similar datasheet from the original manufacturer (Supertex, who was acquired by Microchip) and Vin seems to refer to the output voltage, which is .. weird.
The output voltage is guaranteed to be less than 15V @ 100mA out and your output current will be perhaps 3mA per output. Probably the output voltage will be well under 1V, so allow less than 10mW for all 3 outputs.
There is also a bit of dissipation from leakage currents but that won't normally be significant.
Anytime you consider making a change to something that someone else has spent ... not hours but months and perhaps years making sure it works right ... you need to be prepared to evaluate your change ideas, closely. And that means you need to understand how it works.
Ben Eater has gone to some length to reach out and teach the ideas he's applied, taken to a level almost anyone serious about the topic can follow even without much prior background. In your link to his web page, if you click on the "registers" link there, it will take you to five separate videos that take about an hour to listen to. Just on the topic of his register design.
He starts out talking about using 7474 D-type FFs but then points out that there's a need for tristating buffers, so he trots out the 74245 and a bunch of AND, OR, and NOT gate chips as well, all of which he has discussed in other (older) videos. However, in these videos he's discussing a much simpler approach using the 74173, which is actually designed from the start to operate as a register attached to a bus. It not only includes four D-type FF, but it also includes the AND/OR and enable logic required and it also includes the tri-stating ability for its outputs, as well. So it's pretty much everything all bundled into a nice, easy to use part.
The 74194 does NOT have the tri-stating outputs and it's not designed as a register for a bus, but for an entirely different purpose. You can't just poke around like that. Even if you could get the 74194 to hold values for you, you still wouldn't have the tri-stating capability. And you need that, as well. So you'd now have to change his logic involved in using the part, entirely. It's just wrong on its face.
One thing is sure. You aren't prepared to make your own decisions about swapping parts around like this. So don't.
You already have Peter's recommendation about the two families that kind of "hang together" in acceptable ways. But even then you still need to be wary. Stick with a single family unless you feel competent to do otherwise. Taking into account Peter's answer, you should read at least some of An Introduction to and Comparison of 74HCT TTL Compatible CMOS Logic where you will find the following two tables in "easy to read" comparison format:
HCT was designed to meet the TTL level standards at its inputs and do so with lower input current, but you can see that there are differences in the output voltage and current compliance specifications. It's not likely to bite you. But for example, the HCT output current compliance is symmetrical while the LS isn't. You need to be able to evaluate if this matters in some situation you are considering.
You might want to skim over the above intro-doc I linked above to see if anything catches your attention. Other than that? I'd stick to the plan. There's a lot of work ahead and it will be enough already just getting that much done and working right.
Best Answer
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.)