which user do Azure devops pipelines run as?

I believe the pipeline runs as the "agent". Who the "agent user" is will depend on, first, whether you've chosen a "Microsoft-hosted" or "self-hosted" agent to run your pipeline.

When running pipelines in Azure DevOps that are directly working with Azure Resources you need an Azure Resource Manager service connection. The credentials used to make the service connection are the credentials the pipeline will use when it runs.

You can have your administrator provide you with an Azure AD account that has the permissions you need and then use that Azure AD account to create the service connection for the pipeline. Once you have created the service connection you can use the ResourceID of that connection in place of you azureSubscription.

Here is the link to the Microsoft documentation on creating a service connection.

Here is the link to the Microsoft documentation on the Azure file copy task. To verify you can use the service connection in place of the azureSubscription.

which user do Azure devops pipelines run as?

It depends on the context in which you are discussing the question.

If you mean inside a pure DevOps service, then I can tell you that the user that the pipeline runs on is not the one who triggers the pipeline (which is the case with native DevOps services by default, unless you install some weird extension or have a problem with the pipeline special design), but this identity:

<Project Name> Build Service Account

'run as someone' is just a property of pipeline run. Pipeline run as 'Build Service Account'(On the DevOps side), If you need a pipeline to download or upload an artifact, you can clearly feel this. If the account mentioned above has no permission, you can't do anything.

If you mean the user's operations based on the Azure side, then I can tell you that for native DevOps service, the 'user' that the pipeline performs operations on the Azure side is not the person who triggers the pipeline. The pipeline of DevOps service consists of multiple tasks, and tasks generally interact with other services through service connection (you can find this in Project settings).

There are many types of service connections. When interacting with services in Azure, this type is generally used:

Azure Resource Manager -> Service principal

When you create a service connection of this type on the DevOps side, Azure will also create an AAD app related to this service connection, and this ADD app corresponds to a service principal in Azure. In Azure, permissions are assigned based on service principals or users, and your DevOps pipeline's operations to Azure are based on this service principal and have nothing to do with anything else in DevOps. On the Azure side, this service principal can be considered as an Azure-side 'avatar' of the tasks of the DevOps pipeline.

If you are interacting with Azure through pure code/pure script, then please follow the logic of the script/code.

Our administrator says that the pipeline runs as whoever manually runs the pipeline. If this is true, we would have to give all devs and users read/write access to the blog storage, which would be crazy.

For native DevOps service, of course not. Unless this pipeline has a special design.