WDP dialog is launched from private window

[simonhong]

Description

WDP dialog should not be launched from private window

Steps to Reproduce

1. Launch Brave with clean profile
2. Set Brave as a Private window's default provider
3. Launch private window
4. Create NTP and type search.brave.com
5. Repeat above step 4 3 times

Actual result:

WDP dialog is visible

Expected result:

It should not visible

Reproduces how often:

Brave version (brave://version info)

Version/Channel Information:

• Can you reproduce this issue with the current release?
• Can you reproduce this issue with the beta channel?
• Can you reproduce this issue with the nightly channel?

Other Additional Information:

• Does the issue resolve itself when disabling Brave Shields?
• Does the issue resolve itself when disabling Brave Rewards?
• Is the issue reproducible on the latest version of Chrome?

Miscellaneous Information:

[kjozwiak]

The above will require 1.42.78 or higher for 1.42.x verification 👍

[btlechowski]

Verification passed on

Brave	1.42.78 Chromium: 103.0.5060.134 (Official Build) beta (64-bit)
Revision	8ec6fce403b3feb0869b0732eda8bd95011d333c-refs/branch-heads/5060@{#1262}
OS	Ubuntu 18.04 LTS

Reproduce in 1.42.74

Verified test plan from the description

Set Brave as default search engine for Private Window

Verified WDP dialog is not shown

[LaurenWags]

Updated to release-notes/exclude per discussion with @rebron