This has been a feature request for quite some time now, and we’ve finally got it!  Your administrator will need to enable, “Allow users to relate a contact to multiple accounts” in the org but once that’s done, you’ll be able to add the new related lists on your Accounts & Contacts.

… and yes, this feature is available in Lightning & Classic!

On your contact records you’ll have a new related list for Related Accounts.

The contact’s primary relationship will be specified as direct.  There’s a new objects and page layout to support this new relationship where an administrator can add and remove role options, create custom fields, and set page layouts.

Accounts have a new related list as well, Related Contacts, which show standard contacts, specified as Direct as well as contacts related to the account via a non-direct relationship.

All in all, this will be a nice feature to avoid having to create duplicate contacts in your system if you need to track the same contact related to multiple accounts.