Explaining the Difference
I strongly suspect that the reason an old word was effectively recycled is because the new word filled a lexical gap. It would not really make much sense to use matchmaking and shipping to mean the same thing.
A matchmaker would be somebody who exercises some influence over bringing two people together into an amorous relationship. The active role of a matchmaker bringing characters together makes it practically impossible to be one unless you exist within the same reality or fantasy as people to be matched. The exception is if you analogize the author to a deity, and writes the story as if a match was made in heaven (or heck, depending on the details of the story).
'Shipping, as I understand it as a long standing observer of a few video game and anime fandoms, it is moreso a term of a term of literary analysis used to describe an interpretation of a story that favors certain amorous pairings. It is a largely passive observation regarding interpersonal relations in another. Shipping has no direct effect on how the story turns out being in the end, because you are on the wrong side of the fantasy/reality divide to exercise directly influence the interpersonal relations.
The most active the typical shipper gets is hoping, arguing their case and maybe giving suggestions to the creative staff that can decide whether the relationship will come to pass or not. Only the last of these could be considered matchmaking in any honest capacity. Well, actually they could also create their own unauthorized addition to the story to play out the fantasy somewhat, and they often do. However, that amounts to playing god with alternate continuity versions of the characters than the ones originally 'shipped.
Showing the Difference
Of course, it's easy just to make an assertion, but it is another thing to prove it.
Helping me to do that is to demonstrate something I may have hinted at in my explanation: The lack of influence is not the only difficulty in using the word matchmaking though. The other factor, is that the matchmaker can be a character archetype for people within the same story.
I think the best demonstration of this is to show how the word matchmaker is used in contexts of literary criticism, and the best example I can think of for that is The Parent Trap.
This is a book, famously adapted into a live action technicolor movie by disney in 1961 by Disney which is about twin daughters who seek to rekindle their parents's love for each other for the sake of reuniting their family. The twins are often and accurately described as matchmakers who do matchmaking:
The Twin Test: A Clean Romance by Rule Sinara
"I've seen The Parent Trap.” On DVD, years ago, at her uncle and aunt's house in Nairobi since there was no TV at Busara. “That movie wasn't about individuality. It was about the twins trying to be matchmakers to get their parents back together,"
Twins in Contemporary Literature and Culture: Look Twice
By Juliana De Nooy
And secondly, apart from the occasional comedy of confusion (such as Big Business) and the recurrent 'family entertainment' plot of twin girls matchmaking for their parents (The Parent Trap and its remakes,5 the overwhelming majority of these films are thrillers involving deadly rivalry between good and evil twin sisters.
When people at the bookclub gushes about how Ron Wesley and Herminone Granger from J.K. Rowling's Harry Potter series were always destined to be together, or their peers argue that Harry would have been the better match, that's shipping, as exemplified by the opening line for the Feburary 2nd, 2014 Variety article J.K. Rowling Regrets Pairing Hermione with Ron Instead of Harry Potter by Alex Steadmen
Ron/Hermione shippers, brace yourself: J.K. Rowling thinks the famous pairing was a mistake.
I hope you shall forgive me for not reinforcing this one as much. It is mostly net-slang, and while I could direct you to a variety of discussions where the word is used, such like Final Fantasy Shippers on the Caves of Narshe Forum, it would be difficult for me to source much that is much more authoritative than that, and I also have to tread carefully to avoid hitting disturbingly erotic content that may violate the Code of Conduct.
If you're curious about other uses, you may want to visit some fanfic websites, or maybe not: You see some things may just better left unknown, like how people write about the dysfunctional relationship of Cloud and Sephiroth in the game Final Fantasy VII. Suffice it to say, any place people discuss Japanese pop culture, there will be 'shippers.
A Possible Match
While shipping and matchmaking are both very distinct words, I would suppose that both concepts have considerable overlap with the concept of pairing, which is used in both matchmaking and literary criticism. An example of how the word is used in literary criticism, to establish a an interpersonal character dynamic, can be found in the archive of the May 16th New York Times review Screen: Cruise in Top Gun, by Walter Goodman:
Tom Cruise brings little but a good build to the role of Maverick, and the role of the astrophysicist brings little but impossible lines (''When I first met you, you were larger than life'') to attractive Kelly McGillis. The characters don't pair up well; she's much too classy for him.
While that is not quite exactly the same, the word pairing and shipping are often used in similar contexts. Since we're referencing T.V. Tropes anyway, I might as well share the Crack Pairing:
Shipping beyond all bounds of sanity. There are official couples, there are couples drawn from more-or-less subjective subtext, and there are clearly non-canon pairings that still kind of make sense... and then there are pairings that make you go "What?" Characters who are shipped together despite barely even having a relationship in canon. Sometimes they haven't even met in canon. Sometimes they don't even belong to the same 'verse (that's what Crossovers are for, after all) or to the same species. Sometimes they aren't even both carbon-based lifeforms.
Crack pairings range from the making-you-scratch-your-head-in-puzzlement weird, to the gouging-your-own-eyes-out-with-a-fork weird. But they always leave you asking "Why would anybody think these two belong together?!" [edited emphasis]
The idea there is that you would have to be mentally impaired by drugs in order to support or maybe even think of the 'ship in the first place.
