Search engine for private window switching to standard SE in certain conditions

[samartnik]

Description

Reported by customer https://community.brave.com/t/the-search-engine-for-private-window-keeps-switching-to-brave/536531

Additionally there is UI issue with bottom toolbar showing with wrong color in this case:

Screen_recording_20240425_105936.mp4

Steps to reproduce

1. Set different search engines for private and standard tabs.
2. Fully close the browser (e.g. via exit menu).
3. Long tap on the icon and open private tab.
4. Try to search anything, search engine set for standard tab will be used instead of private.

Actual result

Expected result

Issue reproduces how often

Version/Channel Information:

• Can you reproduce this issue with the current Play Store version?
• Can you reproduce this issue with the current Play Store Beta version?
• Can you reproduce this issue with the current Play Store Nightly version?

Device details

• Install type (ARM, x86):
• Device type (Phone, Tablet, Phablet):
• Android version:

Brave version

Website problems only

• 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?

Additional information

[hffvld]

Verified on Pixel 7 using version(s):

Device/OS: Pixel 7 / panther_beta-user 14 AP31.240426.023.B4 release-keys
Brave build: 1.67.105
Chromium: 126.0.6478.26 (Official Build) beta (64-bit) 

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STEPS:

1. Follow the steps from Search engine for private window switching to standard SE in certain conditions #36758 (comment)
2. Verify

ACTUAL RESULTS:

• Verified that the search results are shown using a correct SE in a Private tab

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2024-06-04_18-05-58.mp4