Pair 2 people based on tiered criteria

Comment first:
Well, as for skill 1 there is just 8 persons with (Y) skill 1, while 22 has no this skill. It's even woese with skill 2: 5/25

But let's try as well as limits allow - I used Solver to assign people to pairs, trying to minimize similarities within pairs.

(if you thing the similarity concept below is misunderstood keep reading to understand general approach, the other understanding of similarity is used in last 2 paragraphs of this post)

And having in mind that we focus on avoiding same skill 1 more than skill 2 and cale least about skill 5 I used as a measure of similarity sum of 1 if skill 5 is the same, 2 if skil 4, 4 if skill 3, 8 if skill 2 and 16 if skill 1 is the same.

Then tried to focus on minimize such situation
Pair 1
N N ...
N N ...

Pair 2
N N ...
Y Y ....

in favor of

Pair 1
N N ...
N Y ...

Pair 2
N N ...
Y N ....

Thus instead of just sums of measures of similarities used sum of squared similarity measures for pairs.

As I said solver was used for it with Evolutionary method and 2 persons IDs being a variable assigned to a pair. The only constrain is that no person could be in two pairs :-) (and additional that person ID has to be larger or equal 1

You may change the starting values in IDs (yellow cells) and re-run solver. The result will be probably different (different pairing), but I expect you will receive rather similar or exactly the same total similarity measure (orange cell, minimizing of it is the goal of the Solver).

(2nd look at what similarity is)

If you literally as you said just try to avoid both members having the same skill (both not having given skill is fine), then the formula calculating similarity (Q3 and down) shall be different:

Formula:

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And Solver is able to generate pairs with all but 4th skill being possessed by not more than one pair member. See sheet 2
Skill4 has Y/Y 23/7 so only 7 pairs can have YN the other 8 will be with both members skilled.