Any issue with using 16 gauge speaker wire for powering LED light strips?
Quality and safety

The cheap Cheese stuff you linked is unspeakably awful, and you really need to stop buying safety related equipment from Amazon Marketplace/eBay/banggood/the other outlets of the Alibaba junkstream.   All of those either involve direct ship from China, or dropshipping via the Amazon warehouse which is effectively the same thing. 

The Amazon Basics product at least reflects some nominal effort by Amazon to deliver a product which meets its product claims, since Amazon is a bricks and mortar direct seller with boots in the USA, making it easy for consumer protection agencies to hold them accountable for their own stuff.  We can't vouch for the insulation quality, but it is unlikely to break down at 24V, because if it did, it would fail its mission as speaker wire.   

If you are confused by the difference between Amazon proper and Amazon Marketplace, that's the plan. 

Anyway, the real question with this AmazonBasics wire is whether it is a type fit for your use, by which I mean visible vs in-wall, outdoor, etc. The cable rating will tell the tale. 

They do make correct cable for such tasks made to go in-wall; thermostat cable comes to mind.  Thermostats are 24VAC RMS(about 35V peak) so that's not a problem. 

Sizes needed for LED lighting

Low voltage LED lighting is a contradiction.  On one hand, you need fairly heavy wire (e.g. 16 AWG) to go any appreciable distance without getting murdered by voltage drop. 

On the other hand, LED strips are so delicate that if you solder stiff, fat 18AWG solid wire to their solder pads, the "tail will wag the dog" as it were - the wire will be so much stiffer than the strip that you are likely to tear the solder pads off the PCB. So you are better off pigtailing off the LED strip with #20 or even #22 stranded wire, then come out to a wire-nut to splice to the long-haul cable.