Suggestions for Matchmaking Algorithm
Although you might find a way to apply machine learning (ML) to this optimisation problem, it does not look necessary, and is probably a distraction. ML might help if the scoring system was complex or if you had incomplete data for most matches, and needed to compute matching score estimates from some more limited set of attributes.

Instead here you seem to have a combinatorial optimisation problem. A well known example of this is the Travelling Salesman Problem. 

There are many possible algorithms to attack these kinds of problem. Which to choose may depend on other traits of the data, such as how quickly you can calculate the scores - both for the whole set and for individual changes. If calculating for changes is fast enough, you can use optimisers that work from a complete (but not yet optimal) solution and make changes.

There is a free PDF/book called Clever Algorithms (Nature-Inspired Programming Recipes) covering selection choice amongst all the varied optimisers. This may allow you to find something optimal in terms of speed and reliability of algorithm.

Here's a simple thing you could try though


Create a "greedy" solution


Shuffle one set of items that need to be paired
For each item in turn, pair it with the best scoring partner
Calculate the score for this solution

Refine the solution


Sample some subset of pairs (e.g. 2, 3, 4, 5 pairs). You could do this deterministically for small enough dataset, or stochastically, or use some algorithm to filter to "at least has some chance of improving".
Find the best score amongst all permutations in this small subset (i.e. brute force all 24 pairings amongst 4 couples)
Put the best subset back into the solution
Repeat until no improvement found after some number of tests



This routine can be altered in various ways to take advantage of aspects of your problem. A nice thing in your case, making this easier that the Travelling Salesman Problem, is that you don't have any constraints on valid pairings. In TSP you care about making a single circuit, and not multiple separate loops which limits how you can make changes, whilst in your case each of your pairs is entirely separate.

One example of possible algorithm improvement: You could pre-calculate the scores for the top N matches for each person, and only search amongst those when considering local changes.