### Apache Airflow version

2.6.3

### What happened

When running multiple standalone DAG processors with separate DAG directories using the `--subdir` argument the processors keep deactivating each other's DAGs (and reactivating their own).

After stepping through the code with a debugger I think the issue is that the calls [here](https://github.com/apache/airflow/blob/2.6.3/airflow/dag_processing/manager.py#L794) and [here](https://github.com/apache/airflow/blob/2.6.3/airflow/dag_processing/manager.py#L798) have no awareness of the DAG directories.

### What you think should happen instead

The DAG processors should not touch each other's DAGs in the metadata DB.

### How to reproduce

Start two or more standalone DAG processors with separate DAG directories and observe (e.g. via the UI) how the list of active DAGs keeps changing constantly.

### Operating System

Linux 94b223524983 6.1.32-0-virt #1-Alpine SMP PREEMPT_DYNAMIC Mon, 05 Jun 2023 09:39:09 +0000 x86_64 x86_64 x86_64 GNU/Linux

### Versions of Apache Airflow Providers

_No response_

### Deployment

Other Docker-based deployment

### Deployment details

_No response_

### Anything else

_No response_

### Are you willing to submit PR?

- [ ] Yes I am willing to submit a PR!

### Code of Conduct

- [X] I agree to follow this project's [Code of Conduct](https://github.com/apache/airflow/blob/main/CODE_OF_CONDUCT.md)

for t in self.tasks:
            t.resolve_template_files()
    def get_template_env(self, *, force_sandboxed: bool = False) -> jinja2.Environment:
        """Build a Jinja2 environment."""
        
        searchpath = [self.folder]
        if self.template_searchpath:
            searchpath += self.template_searchpath
        
        jinja_env_options = {
            "loader": jinja2.FileSystemLoader(searchpath),
            "undefined": self.template_undefined,
            "extensions": ["jinja2.ext.do"],
            "cache_size": 0,
        }
        if self.jinja_environment_kwargs:
            jinja_env_options.update(self.jinja_environment_kwargs)
        env: jinja2.Environment
        if self.render_template_as_native_obj and not force_sandboxed:
            env = airflow.templates.NativeEnvironment(**jinja_env_options)
        else:
            env = airflow.templates.SandboxedEnvironment(**jinja_env_options)
        
        
        if self.user_defined_macros:
            env.globals.update(self.user_defined_macros)
        if self.user_defined_filters:
            env.filters.update(self.user_defined_filters)
        return env
    def set_dependency(self, upstream_task_id, downstream_task_id):