I'm late, but I think some of this wasn't addressed
What exactly does it mean to be "full analog" in games? Does this mean the stick responds differently to a slight movement vs. a larger movement? i.e. not "Switch" activated ON/OFF but rather potentiometer based -- the more you push one way, the greater the impact in the game? I'll have to try the games you mentioned in this thread with the U360's. A related question is the concept of 49-way. What exactly is this? Sorry for the silly questions.
In my books, "full analog" is a game that
needs 256 values per axis (the normal number of values in an analog axis). Games like Term2 need 256x256 grid to hit "everything" on the screen, since the analog x, y value the joystick sends directly maps to a point on the screen; a 7x7 grid wouldn't work, since then you could only hit 49 spots.
"Reduced" analog games can work with smaller numbers, 16x16 or 7x7. These games usually use the analog stick to control a character's direction and speed, and usually only have something like 16 directions and 3 speeds, like food fight or sinistar. A 7x7 grid can cover that. (But foodfight will do better with true analog.)
Which leads to 49-way joysticks/games. They were games designed as "reduced" analog games, to use "reduced" analog sticks. Each axis had 3 optical sensors. If centered, all sensors were blocked, and as the stick is moved from center, sensors are uncovered in an order depending on the direction moved. There are only a few games with the stick, and there actually were two different 49-way sticks. Sinistar, Blaster, Arch Rivals, and Pig Skin are the four with the inputs emulated in mame and used the old (and some say better) "spider" springed sticks. Blitz, Gauntlet Legends, NBA Showtime, and other games on same hardware could use either 8-way sticks, or midway/atari 49-way sticks that you now see at happs. Mame emulates the dipswitch to select between the two. However, mame does not emulate the 49-way inputs of these games ATM. As you might have noticed, I consider 7x7 as "reduced" analog.
Anyway, mame treats 49-way inputs as any other analog stick input, and does any needed conversion inside the game's driver. So, any analog stick works in mame. There are 49-way to USB encoders out there, so the 49-way joysticks work, too. They convert the weird 49-way signal to standard USB analog joystick signal. So these sticks will "work" in mame. But playing term2 with only 49 points to hit sucks.
And last, and least important to most people except wackos like me...

There also were analog sticks that didn't use POTs. The next most common (but still rare), Atari had "Hall Effect" joysticks, and u360 uses that effect, too. Short info: Hall Effect = magnets and sensors that detect the strength of them. But POT based or Hall Effect based, analog is analog. Heck, people have replaced a POT with Hall Effect sensors (flight sim people, mostly, for "better control") without a problem, and vice versa. For mame, don't worry about analog vs Halls vs 49-way; all are analog inputs, and the u360 will work. (It matters if you have the original cab and want to fix it, of course.)