There's no real text book answer for what makes a proper FG, but I will touch on briefly is FG Philosophy. I am going to use SF and MK for this example (if you don't like this, ts)
Essentially, in SF there tends to be a strong emphasis on the neutral game (spacing meta and the SF dance) and controlled offence (you can win the footsie battle but that doesn't mean you have won the match for example)
The pacing is often "slow", methodical and calculated (again, footsies and pokes)
The controls themselves are relatively smooth and the moves have a lot of visibilty to them (start up and active frames are often clearly visible)
It is very easy to reverse engineer the design choices and come up with some logical under pinning for why they are what they are (e,g: Dragon Punch motion starting with a F input creating an element of risk reward; Zangief's Green Hand/Banishing Flat allowing him go phase through projectiles and also giving him mobility; Guile's play style in general and so on)
Compare this to say the MK games where generally speaking they tend to be less controlled (heavy rush down - see chip damage on normals and not just on specials and dial a combos, as well as the flashy nature in general)
It gets a little tricky from this point on as MK games have gradually adopted a more "controlled" perspective to their games.
In any case, there's a lot less deliberateness in the move set in MK games (again, emphasising the flashy natured combat) - try and reverse engineer your favourite characters combos and tools, you will find a number of them serve no real purpose other than padding said characters move set out and/or looking awesome (isn't neccesarily a fault, in fact it is literally part of the design philosophy of MK games and their appeal: FUN)
Similarly, the neutral game works differently with many characters having teleports and/or capture type projectiles (and projectiles themselves not colliding, rather going through each other - iircc this was actually a deliberate design choice by the team way back in MK1)
The dial a combo nature of the combat makes pokes and footsies work differently - your spacing has to account for the dial a combo range. You are by extention rewarded heavily for opening someone up as a result (not so disimilar to SFxT actually).
Also, the controls tend to have this weird I am going to say clunky delay to them which I am not sure is deliberate or not (d,f motions are literally d then f, and not smooth qcf motions for example), there are also no charge based characters (again, highlighting the flashy aggressive nature of the combat but also emphasising the pick up and play factor)
I don't want to bore anyone with too big a post, so I will leave with some homework instead
; reverse engineer Ryu, Zangief and Guile's movesets to get a good understanding of SF's design philosophies and Meta.
As previously mentioned the MK games have evolved so it is a little trickier to find comparable examples, but I would say look at specifically: Sub Zero, and Scorpion for MK design philosophy and Meta. Also, compare how teleports work in MK vs SF.
After that, have a look at this video for an example of complete bullshit character design philosophy...and what to avoid