Yes, that's doable. You'll need to create a CB Auto Action usage to do the storage. Leave Triggers as None and set User to User. If you want to update a field value on click then you can use a Field action for that. Now on the Global tab of your auto action is a URL to directly execute your action. Your button would link to that. To display a button on profile you'll need to use a Custom HTML field and write the HTML for that.

Not entirely sure what you're needing this for though. Could you better describe the purpose or usecase of needing this as there maybe a better or automated way of doing it. For example the CB Last Views Tab already tracks who views your profile.

For that you'll just want to pass that with the URL. When configuring your Custom HTML field add to the URL something like &profile=[user_id] then in your auto action use [get_profile] to substitute it in. Since you're storing this to the database be sure to sanitize it before doing so. So for example you'd be safe to store the following.

[cb:parse function="clean" method="int"][get_profile][/cb:parse]

Be sure to turn on format functions under the parameters tab of your auto action for the above format function to work.