PIP(Picture in Picture) is a Multitasking feature supported by iOS. You can change the subtitle style of the PIP mode in the iOS Settings app.
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Go to Settings app > General > Accessibility.

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Tap Subtitles & Captioning of MEDIA.

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Tap Style.

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Tap Create New Style….

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After you set up a new style, tap Save.

Default

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Apply the Style

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BACKGROUND > Opacity > 0%,
ADVANCED > Text Edge Style > Drop Shadow