I do not think the dies are labelled differently - 5.56 NATO and 223 Remington are like 7.62 NATO and 308 Win, or 7x61 Sharpe and Hart and Super 7x61 - same size on outside of the case, so uses the same die.
For the primers, if you bought cases that use small rifle primers, you are likely okay - most all of the 5.56 and 223 Rem brass here is small rifle primer size. Is possible that there are large rifle primer cases for those rounds - if so, not likely that your small rifle primers will fit.
As you likely know - you will need a source of information - at least one reloading manual - is better to get two or three - tells you HOW to reload - how to trim cases, how to chamfer case mouths, how to set up dies, etc., as much as having pressure tested recipes. Then you will need a supply of primers, powder, bullets and brass cases. Not all of every brand will be the exact same to each other - not all will be the same before and after firing. Most reloaders will prefer to do runs in "batches" - try to keep everything the same in a "batch" - but I am aware of some who could not tell you what brand of brass they just reloaded - let alone how much variance there was in capacity - one to the other. Some folks are happy with a product that goes "bang" - some people want 0.015" 5 shot groups at 100 yards. You will also consume some case lube as you reload - so you will need some of that - many types available.
Fussy people might find a different result with different combinations of brands of brass case, primer, powder and bullet - part of the "hobby" - to discover which combination, of your stuff, that works best in your rifle.
As was discovered when a local fellow showed up to reload a bunch using my tooling - you have to have EVERY part, not just MOST of them - neither he nor I had a shell holder for that case (22 Hornet), for my press - we were stopped cold at that point. We had suitable dies, powder, cases, primers, bullets and tooling, case trimmer, chamfering tool, etc. - everything except for a shell holder of that size.