Kirov as a Soviet status ship

Were Kirov-class battlecruisers designed to counter carriers or Alaska-class cruisers?

No; they were primarily Soviet status units, though their prestige effect helped prompt Iowa reactivation.

Kirov is interpreted as a distinctive Soviet status unit. Its peacetime purpose included influencing perceptions of naval power. The resulting prestige challenge is linked to American reactivation of the Iowa class. Alaska is explicitly rejected as the design's intended opponent.