This particular instance does not involve a semantic specialization of meaning, as the question hypothesized. Instead, it involves two distinct source words with distinct meanings, from different source languages, whose written and oral forms converged from the differences that existed between the source words, until they became identical in the course of their entry into the English language from their source languages.
The two senses of the word "toll" in English are basically different words entirely that just happen to be homonyms that are also spelled the same way. This convergence in spelling is something that happens naturally, whenever homonyms exist, in any language where the written language associated with it employed a basically phonetic spelling system, and was particularly prone to happen in this case because it probably arose before spelling in English had become standardized (which mostly happened in the late 18th and 19th centuries, after the English common law had taken its current form and started using words of Latin origin in their legal sense, like "tolling").
According to Wikitionary, the lay sense of the word "toll" in modern English has an origin in a Germanic root, while the legal sense of the word "toll" in modern English derives from a Latin root (which makes sense because legal language in English, in general, is influenced by Latin to a much greater extent than ordinary English speech, because Latin was a pan-European language of the educated elite in the pertinent time period and because even common law system lawyers in English relied to a significant extent on Roman legal sources, even though they used indigenous sources of law more heavily the Continental Europeans did).
Wikitionary describes the Germanic origins of the ordinary sense of the word as follows:
From Middle English tol, tolle, from Old English tol, toll, toln
(“toll, duty, custom”), from Proto-Germanic *tullō (“what is counted
or told”), from Proto-Indo-European *dol- (“calculation, fraud”)1.
Cognate with Saterland Frisian Tol (“toll”), Dutch tol (“toll”),
German Zoll (“toll, duty, customs”), Danish told (“toll, duty,
tariff”), Swedish tull (“toll, customs”), Icelandic tollur (“toll,
customs”), Latin dolus (“trick, deception”). More at tell, tale.
Alternate etymology derives Old English toll, from Medieval Latin tolōneum,
tolōnium, alteration (due to the Germanic forms above) of Latin telōneum, from Ancient Greek τελώνιον (telṓnion, “toll-house”), from τέλος (télos, “tax”).
It has this to say about the etymology of the legal sense of the word:
From Latin tollō (“to lift up”).
toll (third-person singular simple present tolls, present participle
tolling, simple past and past participle tolled)
(law, obsolete) To take away; to vacate; to annul.
(law) To suspend. The statute of limitations defense was tolled as a
result of the defendant’s wrongful conduct.
It isn't entirely clear whether a third sense of the word, the verb "toll" in the sense of "for whom the bell tolls", derives from the Germanic or the Latin source.
The cognates of the Germanic source words in Latin, are not the same as the Latin word spelled in almost the same way as the Germanic source word which is the Latin root for the legal meaning. The Latin word tollō is a false friend of the Proto-Germanic word tullō; they are not cognates of each other, despite the fact that they have a similar spelling and sound, and despite the fact that both source languages are Indo-European languages.