Are goals simply presets?

Do I lose or gain anything from (dis)allowing my students to have new ones?

Goals are designed to help guide your characters by gaining experience in skills outside of battles and teaching sessions. At the end of each week, each unit will gain a set amount of experience in the skills they have goals in even if they are not promoted to the class that requires those skills.

For example: While a noble, Ferdinand's default skills, strengths, and goals¹ lead him to become a horse-mounted unit such as a Cavalier and Paladin. If the defaults are kept, Ferdinand's riding skill will increase by teaching sessions and goals to allow him to promote to a horse-mounted unit when he reaches the appropriate level.

But what if you don't want Ferdinand to be a horse-mounted unit? What if you prefer armored units? This is where the goals come into play. By changing his goals, you can have his armored skill increase instead of riding to allow him to promote to an armored unit when he reaches the appropriate level.

So by changing (or not changing) your character's goals will effectively change how quickly they can be promoted to a different class.

Is there any significance when they ask me stupid questions where the class name is different but the weapon/goals are the same?

This really all depends on how you want to guide your units, and play style. While the weapon and goal focus may be the same, the class will provide different benefits

For example: The intermediate classes Mercenary and Thief both focus on and require C rank in Swords. However, these classes provide different stat boosts and abilities.

• Mercenaries gain +1 HP, +1 Strength, and +1 Speed.
• Thieves gain +2 Dexterity, +2 Speed, and the ability to open locked doors and chests.

So if your character wants to switch roles to a different class that utilize the same weapon and goals, it doesn't mean they will necessarily mean they will be played the same.

Personally, I like to take the suggested new route whenever the character asks as it generally will benefit them more based on their current skill levels.

If you change their goals when they request, either purposefully or accidentally, you can always change it by back going to Menu > Goals. This will display their default, custom, and requested goals to choose from

Sources

• Kotaku
• Polygon

¹ I can't quite remember the default goals, so humor me :)

Do I lose or gain anything from (dis)allowing my students to have new ones?

Not really, you can change their goals whenever you like, you don't lose or gain anything, they keep the progress of anything they were working on.
For example if you had Caspar working on Swords and Axes, but now want Swords and Lances, he will keep his Axe Experience if later on you decide to change back.

The only thing to watch out for is the characters preferred proficiency. You can simply search up a character on the wiki to quickly see their proficiency, or look them up in game. The proficiency determines how quickly that skill will grow with goals.
For example, Caspar has a good rating for Axes and Fists, but a bad rating for Bows, Reason, and Authority. This means that if his goals are set to Axes, and Bows for example, Axes will train much quicker than Bows, but he will still be gaining Bow skill.

Is there any significance when they ask me stupid questions where the class name is different but the weapon/goals are the same? (E.g. should I master class A axe/brawl or class B axe/brawl?)

Nope! When they come and ask you that, all it does it create a new "quick selection" when selecting their goals. It will not change how fast the skills train or anything else.

For example, I want Caspar to be a knight, so I go into the Goals menu, which would have his starting goal pair, and a selection for custom. I would set his 'Custom Goal' to Axes and Armor.
After a bit, he may come to me and say "Hey Professor, I'd like to learn Axes and Armor to excel at being a knight!", if I accept this, he will have a new preset goal.
So when I go back to into the Goals menu, and choose Caspar, I will now see his starting goal pair, and a new one for Axes and Armor, as well as the custom one, which can still be changed to anything.