From a terminology perspective, there is a Windows Server Failover Cluster (WSFC), and a SQL Server Failover Cluster Instances (FCI). There's also an Availability Group (AG). AGs and FCIs are built on top of a WSFC. AGs and FCIs can be combined or separate. I try to avoid saying "Cluster" and use these acronyms to avoid ambiguity.
From your description, it sounds like you have an AG. An AG is a database-level of HA/DR. Anything and everything within the database is kept in sync between the replicas on your AG. For a visual, look at things in Management Studio, and use Object Explorer to drill-down to see what is contained within the database, and what is outside the database.
sysadmin role is a server role, so therefore that permission is not kept in sync between replicas. Additionally, the server login is also outside the database, and not automatically synced. You will need to create logins and grant server-level permissions separately on every instance that is part of your AG.
Logins, server-level permissions, linked servers, SQL Server Agent Jobs, backup history, (among other things) are all stored in the
msdb system databases, which are not part of the AG, and therefore not kept in sync between instances. You'll need to keep this in mind when you're building a solution using Availability Groups.