Emacs provides a general-purpose highlight mechanism which might work here. See Interactive Highlighting in the Emacs manual.
The relevant commands are highlight-phrase or highlight-regexp to define what you want to highlight (and in what color). You can ask Hi Lock mode to save the highlight instructions in the file itself using hi-lock-write-interactive-...
I know nothing about pdf-tools. But it sounds like you're just asking how to make Isearch's lazy highlighting persist after you stop searching.
If so, the answer to that is to set user option lazy-highlight-cleanup to nil. C-h v tells you:
lazy-highlight-cleanup is a variable defined in isearch.el.
Its value is t
As far as I can see, the problem is that enriched-face-ans does not consider inherited face attributes when it queries attributes with function face-attribute.
It should call face-attribute with the INHERIT argument set to t.
I assume that failing to do so is a bug in enriched-face-ans.
Maybe, you should file a bug report to firstname.lastname@example.org.
Library vline.el (vline.el) does what you're requesting.
If you want the current column to be highlighted as you move around, then turn on vline-mode. (C-u turns it off.) Use vline-global-mode to do this everywhere.
If you instead want a particular column to be highlighted and stay highlighted then use function (not command) vline-show. (Function vline-...