Your world already chose for you, and unless you are trading between worlds you will only ever see one of the two. Cobalt gear comes out slightly weaker across the board but asks for less ore, so a Hardmode world with a thin first ore layer still gets you a full set. Neither option locks you out of anything later; this tier exists to open the next one.

the naming trap

The Cobalt Shield and Ancient Cobalt armor share the word and nothing else. No anvil will make either, and no amount of bars will help. Both are loot only.

where the look comes from

The gear leans hard on East Asian design: a naginata, a katana-shaped sword, a Chinese-styled repeater, headgear built on Japanese and Chinese forms. The metal itself is a dull silver-gray in real life, but its oxide has colored blue glass, glaze and porcelain for thousands of years, which explains both the color and the theme.