What is this military-looking icon count next to my score when waiting for players?

That icon is a Super Sea Snail. You can give them to an NPC in Inkopolis Plaza in order to either add slots to your clothing, or to reroll all three sub abilities on an item that already has all three sub abilities. His name is Spyke and he is located down an alley to your right when you first spawn into the Plaza.

Super Sea Snails were awarded through Splatfests. Just picking a side would award a small amount, while participation would result in being rewarded with even more, based on what rank one achieved and whether their side won the Splatfest. The snails were given out the first time one played after the conclusion of the Splatfest, when the results were presented.

Since we have had the last Splatfest, the method for earning snails has changed. For every 30 online matches won, Judd will give out one snail.

You can get the same effects as you would from a Super Sea Snail by instead spending 30,000 coins, provided you are at least level 20.