matching cartridge and speakers

Interesting question, because you are right, all is a "chain". The X-over from the connected speaker can be a 'black hole' or can serve you more information. When that one is done right it is possible to hear more information, even when the overall gain from your source components is lower...

There aren't any other two components in the audio chain which are more independent from each other than the cartridge and the speaker.
They both are interfaces to the purely electrical part of the audio chain.
Their jobs are entirely in reverse to each other and to choose a cartridge because of a certain speaker or vice verse means nothing than masking apparent faults in either.
There are huge interactions between the output stage of an amplifier and the x-over of a speaker.
This is an electrical interaction (resistance, inductance). Very important to match here.
The cartridge/tonearm is a spring-mass system - a mechanic-dynamic interaction between the two components.
Again - very important to match.
But I see no objective reason to match a cartridge to a given speaker - other than to mask sonic faults and so to fit into a personal sonic matrix.

Dear Inna, this is the most brave thread ever !
So, lets make a start :
If I'm starting from scratch a stereo system, these two would be the most critical to identify it's character and the rest of ingredients (while not of secondary importance) were having the mission to serve these two (not mask their character). Of course we all agree on this.
But what if we proceed by choosing them first?
In this case lets see the process :
1). I would put on my list very carefully those speakers that connects me fluently with music. ie : (Shindo Latour, Red Rose R3) and simultaneously picking the cartridges that also fulfill this exact promise and keeps the spirit intact. ie : (Decca Reference, Colibri)
2). I would choose among my priorities the apparent matching : (The direct liveness of Decca and Shindo, Vs the etheral 3D soundstage of Colibri and R3)

Unfortunatelly or for my good luck, I'm not programmed to make this kind of rationally objective observations a buying decision, once that in reality these theoritical -on paper- mix of equal personas, could give a mega smack in my wallet and bring forth many questions regarding my brain salad. Because the straight wire with gain is a huge lie and everything matters everywhere inside & out of our components especially when they called to mixed together.
Trial & error the only way.
There is not a recipe formula for me. After auditioning all those components in my list, I could not predict the matching pairs among them. Sometimes created an overdose of a particular strength or even a new monstrous character that makes me suspicious for the genuine value of this characteristic strength.

Now in present reality that we allready have choose the synergy between a number of components that as a system satisfy our request, we can proceed to try a new cartridge to pursue our need for pushing the limits higher. Equally we may hear a speaker that could be a new breakthrough discovery to our request and realise that there is no way to reach it's quality of offerings without bring it to our system's chain. If we are not satisfied with our present speaker, there is no catridge that can rescue it from it's misery -"other than to mask sonic faults and so to fit into a personal sonic matrix"- I concur with Daniel.
When I first hear the Shindo 604, I've realised the huge improvement over my PHY Ocellia 03 Silver, but I was also worried about a somewhat heavy, overweight, dull & dark feeling on it's projection at every LP.I recognise that my analogue source certainly could give it a very different personality than Shindo's 301 TT/arm/cart that I was tested it with in the distributor's room. When I've carried my own TT/arm/cart to the auditioning room, the quality went to the stratosphere. But this is my favorite source no matter what the rest items are in the chain and also, I strongly dislike the nostalgic sound of the Shindo's combo. So, we can't say about a good matched source/speaker here. It's just my two favs playing together.

So..one opinion is that there is no reason to match these two; another opinion, that perhaps it could make sense but we wouldn't know how to do it, and yet another one is that we would even be unable to give a more or less clear definition of what to call a match here. And I could guess that someone would say: " If we do this let's also match the table with tubes in power amp ".

Yes, exactly thats the spirit !
(ie) : 300B=sweetness, 211=soundstage, 6c33c=definition
and : BD=lazy, ID=power, DD=flat

BUT : In their best implementation, they must leave no desire for an alternative and this is our will when we choose to buy an item. If we are tolerate and mistakenly accept any wrong of their general family label as an inherent fault, its like we buy something we don't like from start and then we try to fix it with another's item fault. Allways we are in search for the least compromise in everything without any prejudge. If I've heard an amp that I don't like, it does'nt mean the topology or the tube is not to my liking in every other implementation. An example for me was the Jadis Defy 7 a pentode Push-pull capable for greatness in most systems.

When I substitute my DP80 for a Symphonic Line RG6, my Goldmund monoblocks have had a stroke in preservation of timing. After many trials with power amps (PP, SET, OTL) and many other TTs, I've become aware that this SL RG6 was a very bad BD implementation in keeping the timing intact, while the Goldmund power amps love to analyse this in a big way. In such a desperate realisation and trying to save my losted money, I've made more wrong decisions such as pairing this TT with amps that masking its timing problems. Perhaps the outcome as a result in this mix is a great cheater for my audiophile friends, but for me that I've experienced step by step this personal undergoing of adopting & rescue the reputation of this TT, it is an unreversible tragedy knowing the compromise that I accept from the begining. A lesson that I've learn the hard way.

So, yes. " If we do this let's also match the table with tubes in power amp ". The analogy is correct. The results are the same. In the best form of it we can create a cheater monster.
Congratulations for your guts of posting such a topic.