How do I replace this tile/marble transition in a tiled bathroom?
Option 1: Replace with another marble transition

Marble transitions are still sold.  Replacing a broken one with another would preserve the original look of your floor.  However, marble isn't the most durable material to make a transition out of, and it may look out of place if it's the only marble thing in the area.  They're also kind of clunky because of their thickness.

Option 2: Embedded tile transition

There are a number of extruded metal (usually aluminum) transitions designed to be installed in the mortar underneath the tile.  These extrusions can transition between even or uneven flooring heights and present a cleaner, more flush transition between floors.  The caveat here is that the transition placement might be odd, depending on where the wood floor stops in the doorway.

They typically look similar this, but may vary in thickness and profile depending on the height difference:



Option 3: Wood transition

If you can find a wood transition that matches your existing wood (or an unfinished one you can stain to match), you could replace the marble with a wooden transition of similar profile.  The difficulty in matching existing, aged hardwood is not to be underestimated though.  Hardwood tends to darken with age and UV exposure, so a transition that matches fairly well now might not in 5-10 years.  Alternately, you could stain the wood a different color to match wood elsewhere.  Sometimes it's better to go with a completely different color than to get a "close but not quite" match that looks off.

Option 4: No transition

It is possible to butt wood against tile without a transition.  A Google Image search shows many instances of tile being butted against wood without any sort of transitional trim piece, and all of them look pretty good.  This depends on your tile floor being the same height as your wood floor and both floors being cut cleanly and precisely at the transition.  If either of those things isn't true or true enough, I would stay away from this method.  No transition and a rough edge will look sloppy and unfinished.