The big cloud providers have substantial free tiers (50k MAUs). Cognito, Firebase and Azure AD B2C all have a similar free tier. After you go through the free tier, you pay per MAU ($0.005 per MAU, according to https://aws.amazon.com/cognito/pricing/ , I think similar for the other providers. For smaller players, I think Stytch is 10c/MAU/month, where an MAU is any interaction with the Stytch service (login in all the many forms). https://stytch.com/pricing has more.

You'll have to dig a bit deeper into each pricing page to determine what actions make a user 'active'. I'm not aware of anyone who does MAU calculations on a day to day basis; monthly rollups are the standard.

Some folks charge per MAU (FusionAuth does) but in a band (the first 10k MAUs for a FusionAuth paid version are all one price).

HTH.