How to report and assess potential sandboxing circumvention of Apple Music's library on iPhone?

Contact Apple Support, who can read the diagnostic logs on your iPhone for you --> Contact - Official Apple Support

You can also visit your Apple Store Genius Bar to have your diagnostic read.

Do NOT try to read them yourself. They are not human readable as they requires specialized software to be read and interpreted.

Some entity using a sandbox quirk or a sandbox bypass for user tracking would not surprise.

A web browser that has permissions or provisions for access into Apple Music would also not surprise.

Which web browser?

Is that web browser shown in Settings > Privacy & Security > Media & Apple Music > [details]?

As for your question, here is how to: Report a security or privacy vulnerability - Apple Support

I’d not expect an Apple Store to be looking at analytics, telemetry, or details secondary to an privacy report.

Unless using the VPN to connect directly into an associated network or unless you or an associate are running the VPN server yourself, too many of the third-party add-on VPN providers are privacy and security problematic.

Airplane mode is not obviously relevant.

A discussion of performing forensics and of rummaging info.plist and related including the app activity privacy report is going to be a little involved for a forum discussion, though the app and access details posted in this thread are currently also somewhat lacking too.

Post screenshots if you wish as long as they do not contain any personal information to visualize what you are seeing. If you need help understanding the App Privacy Report, you can do that here, contact Apple directly, and even report a privacy vulnerability here.

Report a security or privacy vulnerability - Apple Support

A VPN is actually making your device less secure and Airplane mode is irrelevant when it comes to any data be sent over the internet being intercepted by the VPN provider. It is not a sandbox vulnerability you need to be concerned about there, it is the data that you are inadvertently sending to them. Their security claims are nothing more than a marketing ploy just like Antivirus software that claims to scan your iPhone for viruses. The only purpose is for a "tunnel" to a work/school environment where it is their IT department that provides the access settings to use. A user thinking they are securing their device with a VPN is being mislead and vulnerable.

You haven't actually explained what you are seeing or saw that leads you to believe that anything at all happened...

By the way, unless you're using a VPN to access resources on a private network like your employer or school, it is doing absolutely nothing to protect your privacy or secure your data. Exactly the opposite, actually.