Proceed with Caution, you are entering dangerous territory.
Expanding inspiration in this manner will make it more powerful (by definition). Adding dramatically more options to any ability will do this.
There are also some soft differences between granting success versus granting failure. All party members succeeding once will have a very different feel than a monster who fails four to six consecutive checks.
But these aren't the biggest threat you're going to face. That one belongs to the pure casters...
Save or Suck
While an attack roll on a spell versus a saving throw for its target often seems like an arbitrary distinction, they aren't always. There is a category of spells often referred to as "Save or Suck," which almost universally favor saving throws instead of attack rolls. Spells like Hold Person, Banishment, Feeblemind, Entangle, and Polymorph all inflict near-catastrophic effects on a failed saving throw.
Under your proposed house rule, an optimal tactic would be for players to transfer their inspiration to the full casters in the group (Wizard, Cleric, Druid, Sorcerer), who then uses it to cast repeated save or suck spells on any monster who stands above the crowd.
This has two very negative consequences:
Players are now under social pressure to transfer their inspiration elsewhere, rather than using it to help their own characters shine.
"Boss" fights become incredibly difficult to stage, as any boss has to deal with multiple saves against incapacitating effects with disadvantage applied.
This is not a fatal flaw...
This is not a fatal flaw to the house rule. Monsters written to be boss monsters typically have auto-save abilities, because even without disadvantage save or suck spells are pretty devastating.
In addition, exploiting it requires players to play at a somewhat optimized level. I would not be surprised for an individual group to either not see this tactic, or to be chivalrous about not exploiting it.
But be careful. Make sure that everyone involved is aware that you may back this change out early if it causes problems.
The use of the various types of tools (including the Gaming Set) do not rely on a particular skill but instead simply use an ability score appropriate to what you are trying to achieve.
From the first paragraph on p. 154 of the PHB:
For Example, the DM might ask you to make a Dexterity check to carve a fine detail with your woodcarver's tools, or a Strength check to make something out of particularly hard wood.
Having proficiency with a particular set of tools allows you to add your proficiency bonus to the roll in addition to the bonus (or penalty) granted by the ability score that is being used.
As for how the situation would play out, p. 174 of the PHB describes how to handle a contest between two individuals:
Both participants in a contest make ability checks appropriate to their efforts. They apply all appropriate bonuses and penalties, but instead of comparing the total to a DC, they compare the totals of their two checks. The participant with the higher check total wins the contest. That character or monster either succeeds at the action or prevents the other one from succeeding.
If the contest results in a tie, the situation remains the same as it was before the contest.
In your particular case it would depend in part on what game they are playing and what sort of approach they are taking. They could maybe use Charisma to try and bluff their opponent. Or they could try and read their opponent with an Intelligence or Wisdom check. Maybe even Dexterity to try and discreetly slip a winning card into their hand.
Participants do not have to use the same ability check: the NPC could try to bluff the PC with a Charisma check while the PC tries to read the NPC with Intelligence.
Best Answer
No, not when used properly.
DM has control over what inspiration encourages
The key thing here is that the DM controls when inspiration is awarded. As a DM, I give it freely and often and have never had a case of My Guy Syndrome (MGS) come specifically from this mechanic. In fact, I use it (successfully) as a way to reward players for non-MGS behaviors (behaviors the group as a whole find fun).
It could be used to create a worse problem though if you encourage the wrong things. In this case, you can make an MGS issue worse if you keep rewarding actions that are negatively affecting group fun in the name of good character RP. However, I think inspiration can only amplify a problem that already exists.
I sincerely doubt, and have never observed, that inspiration as a mechanic is powerful enough to actually cause MGS behaviors to appear outside of any other factors. And, on the flip side, when I have had problems with MGS at my table, inspiration has never been enough to fix it (and also runs afoul of my policy to fix out-of-game issues with out-of-game solutions as a first step).
My Guy Syndrome vs. Roleplaying
You seem to be conflating two things: role-playing and My Guy Syndrome. Role-playing often involves having a character have some sort of flaws or acting out drawbacks related to their personality or backstory. That's all part of storytelling. If every character was perfect and unflawed, stories about characters overcoming those flaws or developing as a character cease to exist. This is the kind of thing this passage is providing a mechanic to encourage.
On the other hand, My Guy Syndrome is what happens when you role-play any part of your character in a way that negatively affects your table's enjoyment and fun and then justify it as the only way you can act because it is what your character would do and that you have no choice but to do this.
If you are acting out a character flaw, and there is no issue with how much fun it is at the table and is not causing any other issues, then there is no problem and thus no MGS.
Other parts of the book provide more guidance that should be used with this
MGS is something that only occurs when a player is roleplaying as their character problematically. Anything that encourages more roleplaying without explicitly telling the player not to do so could be construed as "encouraging" MGS. However, the rules must be read in context. The portion that you read is in the DM's guidance for inspiration. In the PHB it says:
Not the emphasis on group fun and story? That is pushing the players away from MGS. Taking one section of one part of the DM's manual without any of the other guidance the books offer on the matter is bound to give a skewed view of the guidance it offers. As with most things: context matters.