Anthropic has an API, you can use any client but they charge per input/output/cache token.

One-price-per-month subscriptions (Claude Code Pro/MAX @ $20/$100/$200 a month) use a different authentication mechanism, OAUTH. The useful difference is you get a lot more inference than you can for the same cost using the API but they require you to use Claude Code as a client.

Some clients have made it simple to use your subscription key with them and they are getting cease and desist letters.