Notepad3 (x64) v5.21.218.2 rc
Windows 10 Enterprise 20H2 x64
When highlighting a selection that begins with a "$", the Occurrences element on the Status Bar does not update:

It does update when opening the Find or Find/Replace dialogue:

...but the occurrences reverts to the original behaviour when the Find(/Replace) dialogue is closed.
NB: this behaviour is not new; it happens as far back as version 5.20.915.1 (the earliest version I have at hand).
As a related aside, double-clicking on the string results in a selection that excludes the "$", and the Occurrences element reads correctly:

Should a double-click exclude the leading "$"? Is there a setting that when changed would lead it to be included?
This is caused by Mark Occurrences mode Match Whole Word Only for the editor view.
The '$' sign is not part of a normal word (test: double-click the word, the '$' sign is not selected), so the complete selection does not fit the criteria of being a word to find occurrences for Match Whole Word Only. 鉁旓笍
While in F/R-Dialog mode, the Match Whole Word Only the Mark Occurrences mode (if checked in the dialog) is switched to mode Match Search Pattern (according to selected options), which now finds your occurrences. 鉁旓笍
You can switch to Accelerated Word Navigation (Menu->Settings->Accelerated Word Navigation Ctrl+Alt+A), where the only word separator is space (blank or tab or line-break). This will recognize your complete expression as a "word"and find your occurrences in Match Whole Word Only mode. 鉁旓笍
Additional word separators (additional to blank, tab or line-break) can be defined in[Settings2] ExtendedWhiteSpaceChars=... for the word separation in Accelerated Word Navigation mode.
Unfortunately, regarding your last question, you would need the "reverse" setting (a la _ExtendedWordCharacter=_) and add the '$' sign to it. We don't have this settings option. Maybe you can live with defining some usual word separators like !",;:/|\+*=搂&()[]{}# etc. , for Accelerated Word Navigation 馃
Or you can switch off Match Whole Word Only mode (Menu->View->Mark Occurrences->Match Whole Word Only->OFF), which activates a mode similar to the F/R-Dialog mode based on current selection. 鉁旓笍
This can be a bid annoying, cause even a selection of a single character triggers the occurrences marking for the complete document.
Thanks... Still digesting this and figuring out the best way forward for me.
Most helpful comment
Thanks... Still digesting this and figuring out the best way forward for me.