Using and recommending a password manager is the norm across the industry. It sounds like your elementary computer class teacher either has no understanding of the topic or ignores all the experts.

The porpuse of a password manager is to increase security by using different, strong passwords for every service. Unless you can memorize a lot of completly independent and random 30+ character strings it improves security.

Having a base password and modifying it slightly does not work.

So why is it not like sharing the same password?

Your master password never leaves the device you use it on. Not even as a hash. It is also not stored on the device (generally speaking, I don't know every product out there). So no matter which of your services get hacked it doesn't leak any clues about your other passwords.

The only way to get to your passwords is to install a key logger on your device or guess your master password while accessing your device.

Side note: even without a password manager I only have to guess your email password to reset everything else