So We Call It Metroidvania
The name of this genre originated from the NES game Metroid and the Playstation game Castlevania: Symphony of the Night. By taking the name of the first game and the removing the castle part from the latter we get our genre name. Similar to how the name of this genre is created the genre categorizes games.
Features In Metroid
Metroid games are jump 'n' run games that include inspiration from adventure games. They consist of a big interconnected world with abilities and power-ups or upgrades as well as bosses lying around. The abilities, power-ups, and upgrades allow you to progress further in the game and it's not uncommon that you need to backtrack. Additionally, bosses and standard enemies require you to have certain abilities to defeat them easier or at all.
Features In Castlevania
Castlevania: Symphony of the Night has you explore Dracula's castle, a big interconnected world as well. There are items, power-ups, upgrades, and abilities here as well, same with bosses. The upgrades and abilities are required to progress further in the game as well. Backtracking is part of this game as well. So, basically, this version of Castlevania is the same as Metroid... except there's more. Compared to Metroid, Castlevania includes RPG elements, such as stats and leveling.
The Fusion
So, the combination of both, the Metroidvania, is a jump 'n' run adventure game that consists of big interconnected worlds that allow you to semi-freely explore it, but require you to backtrack once you get a new ability to find new paths or secrets that will lead to new abilities. Additionally, several bosses are scattered around that either halt your progress by being in your way or defeating them gives you an item that is required.Modern Game Example
One of the modern Metroidvanias is Hollow Knight. It was nominated four times and its rating is between 86%-100%, depending on the reviewer. Hollow Knight, like the other two games mentioned here, uses a big interconnected world map with abilities, power-ups, and bosses scattered around. Backtracking is not uncommon in Hollow Knight either and while none of the abilities are required for the bosses they do indeed make it easier. Compared to it's Castlevania counterpart it does not contain RPG elements such as leveling and multiple stats though.Is Banjo & Tooie a Metroidvania or a Collect-a-thon?
When searching for 3D Metroidvanias (since typically Metroidvanias are 2D games) I came upon a post where people called Banjo & Tooie a 3D Metroidvania. With the earlier post about Collect-a-thons and this post about Metroidvanias, we should be able to answer that question.
Does It Have a Big Connected Map?
Banjo & Tooie has multiple hub worlds that are connected with each other and they are connected to single levels with each having its own theme. These single levels are however partially connected to each other as well. Now it depends on where you draw the line for big interconnected worlds since there is a difference here compared to the other games I've mentioned.Does It Contain Backtracking?
This question I can definitely answer with a yes. Something I loved about Banjo & Kazooie was, if you go into a world you can 100% it. This is not the case in Banjo & Tooie. There were collectibles you couldn't reach without having been in another world or having unlocked a certain ability.Are Abilities Scattered Throughout the World With Different Means?
What I mean by this is... if you can buy some abilities, find some lying around, find some in a chest or get some from a boss then this matches the other Metroidvanias. This is not the case in Banjo & Tooie all abilities are unlocked by buying or using collectibles. Which makes sense for Collect-a-thons.
What's the Primary Objective For Progress?
Metroidvanias primary objective consists of challenging the player having him rise with the challenge and explore the world. Collect-a-thons sole purpose consists of finding and collecting objects and progress is tied to it. Now, of course, you might end up exploring the world in a Collect-a-thon but the purpose is different. You're exploring in hope to find some kind of collectible. In Metroidvanias you explore to find new paths and abilities. In Banjo & Tooie the only way to progress through the game is by getting jigsaw pieces to unlock new levels to get more jigsaw pieces to get more levels until the last boss.
Conclusion
As I mentioned whether or not you consider Banjo & Tooie as a Metroidvania depends on where you draw the line, however going through these questions I personally would say, it is not.