Unless I've misunderstood you, as your question isn't 100% clear...

After, for example, cookie reset I need to log in again.

This is default behavior for all websites which use a cookie based authentication system.
If the Session is reset somehow, including deleting cookies, then you are logged out.

Why for N StackExchange sites I need to enter the password N times? Isn't the account consoladated?

You shouldn't need to.
  If you are logged in to one Stack site and visit another which you have already associated your account with, then you should either:

1. Be auto logged in when you visit, and just click the refresh in the top banner
2. Sometimes happens when you need to manually click "login", but then you should only need to click the login method after that and the site will get your cookie from the other site you are logged in from

Although this does depend on your login method. Can you give more info?

Sometimes I can wait for some time and refresh the page and it says at a top something like "Oh, it's you. Click here for the login.", but not in this case

Did you clear cookies? Which would log you out of all sites.

Can you describe the actual issue in a little more detail?

Unless I've misunderstood you, as your question isn't 100% clear...

After, for example, cookie reset I need to log in again.

This is default behavior for all websites which use a cookie based authentication system. If the session is reset somehow, including deleting cookies, then you are logged out.

Why do I need to enter the password N times for N Stack Exchange sites? Isn't the account consolidated?

You shouldn't need to. 
If you are logged in to one Stack Exchange site and visit another which you have already associated your account with, then you should either:

1. Be auto logged in when you visit, and just click the refresh in the top banner
2. Sometimes happens when you need to manually click "login", but then you should only need to click the login method after that and the site will get your cookie from the other site you are logged in from

Although this does depend on your login method. Can you give more information?

Sometimes I can wait for some time and refresh the page and it says at a top something like "Oh, it's you. Click here for the login.", but not in this case

Did you clear cookies? Which would log you out of all sites.

Can you describe the actual issue in a little more detail?

Unless I've misunderstood you, as your question isn't 100% clear...

After, for example, cookie reset I need to log in again.

This is default behaviour for all websites which use a cookie based authentification system.
If the Session is reset somehow, including deleting cookies, then you are logged out.

Why for N StackExchange sites I need to enter the password N times? Isn't the account consoladated?

You shoulnd't need to.
If you are logged in to one Stack site and visit another which you have already associated your account with, then you should either:

1. Be auto logged in when you visit, and just click the refresh in the top banner
2. Sometimes happens when you need to manually click "login", but then you should only need to click the login method after that and the site will get your cookie from the other site you are logged in from

Although this does depend on your login method. Can you give more info?

Sometimes I can wait for some time and refresh the page and it says at a top something like "Oh, it's you. Click here for the login.", but not in this case

Did you clear cookies? Which would log you out of all sites.

Can you describe the actual issue in a little more detail?

Unless I've misunderstood you, as your question isn't 100% clear...

After, for example, cookie reset I need to log in again.

This is default behavior for all websites which use a cookie based authentication system.
If the Session is reset somehow, including deleting cookies, then you are logged out.

Why for N StackExchange sites I need to enter the password N times? Isn't the account consoladated?

You shouldn't need to.
If you are logged in to one Stack site and visit another which you have already associated your account with, then you should either:

1. Be auto logged in when you visit, and just click the refresh in the top banner
2. Sometimes happens when you need to manually click "login", but then you should only need to click the login method after that and the site will get your cookie from the other site you are logged in from

Although this does depend on your login method. Can you give more info?

Sometimes I can wait for some time and refresh the page and it says at a top something like "Oh, it's you. Click here for the login.", but not in this case

Did you clear cookies? Which would log you out of all sites.

Can you describe the actual issue in a little more detail?