OK, so we need to know which one of the "Two's" you are using. On your Mac, go to: Apple - My Apple ID, sign in with your @me.com Apple ID, and under the Security section see whether it says you are using Two-Step or Two-Factor.
I'm a bit confused by where you are trying to sign in your @me.com email address. If you are trying to sign into Gmail, wouldn't it be an @gmail.com email address? (sorry, I don't have an Android, so not sure where the @me.com address comes in here).
The answers to those questions will help to establish what your setup is, so if there are other issue, we will know what base we are working from.
As for getting signed into an app on your Android that will be accessing iCloud data, you need to sign into that app with your Apple ID and an App-specific password. Your Apple ID password does not come into play for this sign in.
On the same Apple ID page that I provided the link for above, you will also see in the bottom right-hand corner of the Security Section a label with "App Specific Passwords", and below that "Generate Password". Type in a name you want to categorize the password with - i.e., Android Mail Password (or anything you like):

Once you have created a label, then click on Create. You will get a long password with 4 sets of 4 characters separated by dashes. This is what you will type into your password field. Nothing else - only the App-specific password.
A few years ago, even Apple devices were required to use App-specific passwords when signing into Messages or FaceTime. I could use the same password for signing in both on as many devices as needed. If you get signed out of the app and need to sign in again, you can create a new App-specific password if that is easier than keeping the original one somewhere. Since I rarely got signed out, that is what I would do if I was somehow signed out.
Does that help?
GB