Rework the logging module.

This is a summary of various issues related to the existing logging module.
- It's a thread local variable, which runs into issues with logging in child threads and distributed systems. https://github.com/dmlc/xgboost/issues/9147
- It creates symbol conflicts. https://github.com/dmlc/xgboost/issues/6678 https://github.com/dmlc/xgboost/issues/10400
- It doesn't translate into binding-specific logging facility very well. For example, how to use Python logging with file output?
- It should have a separate channel/logger for different types of logs. For instance, the memory allocation log should not be mixed into other debug logs.
- Backtrace should be optional. https://github.com/dmlc/xgboost/issues/11758#issuecomment-3423228972
- It should print XGBoost version: https://github.com/dmlc/xgboost/issues/9914