USB C or other compact connector for 19 identical, general purpose lines?
19 you say?

By coincidence, HDMI does have exactly nineteen conductors.

But: three of them are shields, some are differential pairs, which are optimized for high-frequency signals instead of low-frequency current transport… I doubt a single one of them was designed to carry current for a 20 mA LED, leave alone a "power LED" (whatever that is).

In short: all these modern computer cables are specialty devices and you can't expect them to just be "a bunch of wires".

So, stick with something else. SUB-D connectors are pretty standard, and they are easy to deal with with hand tools; of course, they are a lot larger. But you might need that anyway, because "power LED" kind of says each conductor might carry 1 A or more, and that's not going to be fun with smaller connectors...

If you asked me, this sounds like an architectural thing: Instead of buying (expensively) copper to bridge a large distance with a lot of current (losing power to wire resistance), you should probably have your (or another) Arduino where you'd  have the other end of that cable you're wishing for, and just run a data line and power there.

In fact, there's really easy ways to switch a lot of outputs with few cables: 

For example, there's linear shift registers with a latch input; there's even some that come with integrated linear LED drivers with adjustable current. With these, you'd only have to run 5 signals


power
Ground
Data
CLK
Latch (optional)


That sounds a lot easier and less error-prone then having 19 conductors! Also, you're not limited to 15 LEDs that way. You can have as many as you want.