Capture if the user is in private browsing mode

### What is this about?

Capture this property as part of a sentry error, may help investigating the issues as we've had issues with PB before

### Scenario

_No response_

### Design

_No response_

### Technical Details

_No response_

### Threat Modeling Framework

_No response_

### Acceptance Criteria

_No response_

### Stakeholder review needed before the work gets merged

- [ ] Engineering (needed in most cases)
- [ ] Design
- [ ] Product
- [ ] QA (automation tests are required to pass before merging PRs but not all changes are covered by automation tests - please review if QA is needed beyond automation tests)
- [ ] Security
- [ ] Legal
- [ ] Marketing
- [ ] Management (please specify)
- [ ] Other (please specify)

### References

_No response_

The issue does not name files, tests, or an entry point. Start by locating the TypeScript Sentry error-reporting path and reviewing how browser private browsing mode can be detected across the supported extension browsers. Done means the private-browsing property is captured with sent errors and the relevant behavior is covered by tests.

Written by the indexing model from the issue text.