Christina Edkins was killed on a double-decker in Hagley Road, Birmingham, during the morning rush-hour on Thursday.
A spokesman for West Midlands Police said the man will now be sent to a secure facility after the latest assessment by mental health professionals deemed him unfit to be interviewed any further by detectives.
Police said the suspect was arrested near the scene of the attack.
Christina had been sitting in front of her attacker on the top deck when he struck as the bus was slowing down.
Post-mortem tests have revealed that the teenager died from a single stab wound to the chest.
Detective Superintendent Richard Baker said after the attack: “He walked down the stairs and calmly waited to get off the bus.”
This afternoon, friends of the teenager gathered to pay their respects at the scene, where a growing number of floral tributes have been placed.
Police have released a picture of the suspected killer, a black man in his early 20s, who was wearing a distinctive jacket with a picture of a tiger on the back.
They believe he used buses in and around Birmingham around the time of the attack and have appealed for anyone who saw him between 6am and 1pm to come forward.
They also called on any remaining passengers from the number nine bus to speak to them about what happened.
Mr Baker said: “We are appealing for people on the bus at the time of the incident to please come forward.
“We know from witness accounts that, although some screaming was heard immediately after the incident, initially people thought perhaps it was just teenagers messing around on the top deck of the bus.”
The arrested man, a British national with links to Walsall, was not known to mental health units in the area and has no known link to the teenager.
Police said it was part of Christina’s regular daily routine to get the bus at that time.
Mr Baker said the teenager’s family were distraught over the apparently motiveless attack.
He added: “Clearly they are absolutely devastated by what has happened. If you look at the pictures of Christina, she was a beautiful young girl. This attack appears to be totally senseless and random.
“The dedicated officers in the investigation team are working hard now to make sure the person responsible for this is brought to justice, and we will continue to work closely with the Crown Prosecution Service over the coming hours in order to make that happen.”
A woman claiming to have seen the incident said she waited for the attacker to get off the bus before performing CPR on Christina.
Writing on Facebook, Stephanie Bent said: “I was on that bus I knew what he had done and he was stood next to me when he was getting off, all I could think of was I hope he gets off this stop so I can go upstairs and help the girl.
“Thankfully he got off and I was able to go upstairs and help the girl. I gave CPR until medics arrived and am deeply traumatised I could not save her, I certainly wasn’t going to confront a knifeman knowing he had just stabbed someone I thought he may stab me too all I cared about was getting up to the girl to help.”