Sadly standard is a thing that exists on PCs, some cars (though even that is variable and I have bought whole old harnesses at considerable expense to get connectors before, and regularly wholly replace things with others, and even things that are seriously standardised like radios I still have to adapt a lot of the time) and within device classes (find a dead one and rip it off that*). You might get lucky and find one, someone that knows the flavour and can direct you or the like but there is no list of components we share around for repairing things in circles like this.
*there are those that will buy up cheap versions at yard sales, auctions, flea markets, car boot sales, strip components off them and sell them on individually (individually worth less than the sum of its parts) on the usual places for second hand tat. I tend not to rate the chances of most plastic connectors being desoldered mind you, though if they are willing to stake a name on it then if it works then it works (40% success of having a viable component when you desolder it is still 40% so get a stack of cheap devices, strip them all and you will probably get something).
Connectors (see also interconnects, terminals and sockets) are largely sold at the same places that sell electronics components if you did want to try. There are various companies that make them, and families/classes they make individually so have fun there.
This also isn't like amazon or ebay and there are usually minimum spends, or minium spends to get free shipping (else it is small fortune) which is rather at odds with the 2 cent connector you might find for this. If you are at a school, university, know a local electronics repair shop/manufacturer then they probably have orders all the time with them and are usually happy enough to tack on something more and call you when it gets in if you go ask nicely. Pro tip is when it is a 2c part order 10 of them or whatever you would not stop to pick up a coin for, not so bad here but just in case you do melt it or sneeze and it goes flying then a replacement on hand is nice to have.
https://www.digikey.com/en/products/category/connectors-interconnects/20 (this is probably the biggest vendor for most of the world and almost certainly US)
https://www.newark.com/c/connectors...-INTERCONNECT_PCB_RECEPTACLES-MAY20-WF1980967
https://www.mouser.com/c/connectors/
https://www.sparkfun.com/categories...ory_91&filter_price_floor=&filter_price_ceil=
https://www.avnet.com/shop/emea/c/connectors/
https://export.rsdelivers.com/category/electronics-components-power-connectors/connectors
They will all tend to offer something called a parametric search which you can narrow things down with, as opposed to being all about being a catalogue like other selling sites.
About the only difference between a conventional mechanical connector and this is you might have to specify number of pins and pin pitch (as in distance between them). Everything else is defined by standard mechanical engineering drawing, and maybe skip that for a brand if they have certain families with similar features. Nintendo could have used a standard type but whether it is still sold (we are now however many years on, and like most plastic components they tend to get cycled out leaving you hunting for old stock**) and what that is I tend not to see, and I don't think the gigaleaks included a nice bill of materials or schematics that include vendor names, numbers and the like.
One note with connectors is they might not ship with easy to solder connectors on (such things tend to make for larger components than other methods) and thus you get into specialist crimp tools, needing ferrules and the like. You should be able to tell from the listing/datasheet, and if not hopefully you get lucky and it is a push connector you can put the ferrules on and slide into place with.
**which is a whole other field unto itself, normally more in manufacture settings but repair comes into it too.
https://octopart.com/electronic-parts/connectors
https://www.findchips.com/parametric/Connectors
https://www.alldatasheet.com/
Those being more where you go to find such things, and whatever vendor still has stock of them, though the above might also do and have some legacy stock of things.
If playing cowboy then do also bear in mind you only care about the pin pitch, suitable male/female, and number of them (at least have enough, removing/ignoring redundant is possible and maybe even preferable in some scenarios). Can happily get something from a dead device or from a cheapy shop that joins to the relevant pins and either file down excess (including alignment/this way up stupid pins that might be on top) or make up difference with some form of glue (polymer adhesives, think like caulk but more for this, being more what you will want to look at than superglue or epoxy) so it does not wiggle out.
This is assuming you don't do what the OP eventually did and just solder wires across instead (quick, easy if you know how, cheap and reliable)