Yes, good web app design never trusts the input from the client; always assume that it could be malicious.
After authentication and that means that he has access to the API,is it possible to alter the request somehow like with Burp suite to send an other Id to the View in order to make the API malfunction and return another user's details?
Whether that will cause your app to malfunction and return data that it shouldn't will depend on whether the app is well built or not :)
In other words, if someone authenticates, thus a legal user, how do you protect the API from him forging the requests, Post or otherwise, to gain further access or retrieve results of another user by sending parameters that will be fed to SQL?
You need to have an authorization layer in the API server. For each request you need to retrieve the userID from the auth token and ask "Is this user allowed to perform this query?". To make it even more complicated, sometimes they are allowed to call that endpoint, but should only see part of the results; for example, maybe every user is allowed to call
GET /api/users, and maybe a regular user only sees themself, a group admin sees all users in their group, and a super admin sees all users.
There's no magic or secret formula here, just good API design and coding practices, like using different DB queries / filters depending on the user's type.