This is a bit difficult to get your head around but is very doable. It's kind of hard to explain so forgive me if this is difficult to follow - I'll try to make sense!!
First of all, you'll need to have the player collision set up in the right way. For top-down games, you should ideally have the player's collision object set to be a small shape at the character's feet (like you are controlling their 'shadow'). This will allow you to easily keep track of where the character's feet should be in relation to the ground.
"Jumping" in a top-down game is kind of weird to get your head around, but really, the player collision object doesn't jump - only their visual representation. In the first Startropics screenshot you posted, you can imagine that what you are really controlling is the guy's shadow, and the actual character sprite just moves up and down to look like a 'jump'.
So, to implement jumping all you need to do is:
1) Give the character an animation that looks like 'jumping'
2) Add obstacles that the player's collision object cannot pass unless the character is in their 'jumping' state
Is this making sense? 🤔