Why is Marcus Antonius’ name anglicized to “Mark Antony” while other contemporary politicians are still the same. ie Gaius Julius Caesar, Lepidus etc

Some combination of Shakespeare, convention, and French.

Speakers and readers of English have been engaging with the Classical world for a very long time, and a few prominent figures (like our abbreviated friend Mark Antony) have had illustrious independent careers in English literature and cultural reference. In Antony's case, the definitive English-language touchstone is of course Shakespeare's Antony and Cleopatra, in which, as the title advertises, Antonius appears throughout as Antony. The cultural prestige of Shakespeare is probably responsible for making Mark Antony the standard form of the Triumvir's name in English.

Why did Shakespeare use Antony instead of Antonius? His principal source for the play, Thomas North's translation of Plutarch, does not shorten Antonius. Shakespeare used "Antony," I suspect, partly because "Antony" makes a much nicer ending to an iambic pentameter line than "Antonius." Lines like this:

Sir, sometimes, when he is not Antony,

He comes too short of that great property

Which still should go with Antony.

Would sound half so sweet with Antonius in 'em.

In choosing to use "Antony," Shakespeare was also following an existing English convention. In 1594, for example, more than a decade before Antony and Cleopatra appeared, Shakespeare's contemporary Samuel Daniel composed a(n unreadable) Tragedy of Cleopatra, in which Antonius appears as Antony. Daniel, in turn, followed the lead of his patroness the Countess of Pembroke, who had translated the French tragedian Robert Garnier's Marc Antoine as The Tragedie of Antonie. So the ultimate culprit may well be the French habit of making Marcus Antonius into Monsieur Marc Antoine.