"Fair share of taxes" seems like it should be pretty self-explanatory.
Assuming one isn't an anarchist or communist, the goal of structuring taxes "fairly" seems pretty simple. Either...
The amount one pays is as close as possible to the government benefits they receive, or
Everybody pays more or less the same amount across the board
Like... let's look at payment close to benefits received first because it seems the fairest. The thought behind gas taxes is that the people using the road most are the ones who pay for road maintenance. EVs might complicate this, but it's an attempt, and my state tries to compensate by having a higher registration fee for EVs which is just another kind of use tax. The thought behind FICA funding entitlements is that everyone pays some, and it's at least loosely linked to benefits received because payments are tied to contributions even if it's not 1 v. 1.
It makes sense that it can never be exact, and it even makes sense that there will be some small amount on top of whatever the system is because perfect fairness isn't possible. Like... we all supposedly benefit more or less equally from, say, national defense, or having adequate levels of education or sanitation in our towns or whatnot. But the easiest way to get closer to a fee based structure similar to how one pays for services, the more fair it is.
The other way is everyone pays something equal. Some people would use that as an argument for a flat tax, but if it's a percent of income, that's not equal at all and the rich are paying a disproportionate amount. A better example might be stuff like excise taxes on commonly bought goods. Like... a sales tax still means that the rich are paying more because they're spending significantly more, but it's at least a little more proportional. And as an added benefit, when you compare it to an income tax? It taxes consumption rather than productivity, at a time most people think we're consuming too much and not producing enough.
In essence, those are the only two ways it makes sense to read "fair share of taxes", and the people who use the phrase the most never seem to define the term either rational way, or really any way at all. What's really bonkers is that, despite dismissing both readings that seem to be the only obvious ones, they expect you to know what they mean without them having to define their term. Sticks in my craw every time, like they're living in an entirely different world and speaking an entirely different language.