There is a contact form in the contact page, which is perfectly working fine. I want to add the same form in a different page (say product page). How can I do that in Drupal 8?
You could use Contact Block module which will allow you to place a contact form block on whatever page and region you desire.
Once installed place a block as usual and filter for contact block, then in the block configuration settings you can choose which contact form to use. You can also add pages to display the block, remember to add pages with a slash "/my-pages-path"