"It's possible that the person who was supposed to do this was [a coward] because the system was intact," he said.
One blast hit an Ethiopian restaurant in the south of the city on Sunday, while the other occurred at a rugby sports club as people watched the World Cup final.
The near-simultaneous attacks on Sunday were the first time the group, which has carried out multiple suicide attacks inside Somalia, has struck outside of the country.
"Al-Shabab was behind the two blasts in Uganda," Sheik Ali Mohamud Rage, the group's spokesperson, announced in Mogadishu.
"We thank the mujahideens that carried out the attack. We are sending a message to Uganda and Burundi, if they do not take out their Amisom [African Union Mission in Somalia] troops from Somalia, blasts will continue and it will happen."
Uganda and Burundi currently have peacekeepers in Somalia as part of a stabilisation mission supported by the African Union.
"[Al-Shabab's] strategy is to undermine getting troops into Somalia through attacks like this," Simmons said.
Hussein Mohammed Noor, a Somalia analyst, said the Ethiopian restaurant was likely targeted because of "Ethiopia's involvement in Somalia".
However, he told Al Jazeera that these attacks were unlikely to make African countries reconsider sending troops to Somalia.
Lieutenant-Colonel Felix Kulaigye, a Ugandan army spokesman, said: "Al-Shabab is the reason why we should stay in Somalia. We have to pacify Somalia."
Kayihura that the attacks, which took place amid large crowds at the two locations, could have been carried out by suicide bombers.
"These bombs were definitely targeting World Cup crowds," he said.
Severed head found
Investigators reportedly found the severed head of a Somali national at the scene of one of the blasts.
Officials said 60 Ugandans, nine Ethiopians or Eritreans, one Irish woman, and one Asian were also among those killed.
Two people could not be identified. At least 85 people were wounded.
The attacks left scores of football fans reeling in shock.
"We were watching soccer here and then when there were three minutes to the end of the match an explosion came ... and it was so loud," Juma Seiko, who was at the Kampala Rugby Club, said.
Hassan Isilow, a Somali analyst living in Kampala, said that Somalis in Uganda feared reprisals after the claims that al-Shabab launched the attacks.
"There is fear within the Somali community at the moment," he said. "People are in panic."
"[Somalis] own lots of businesses around the city and most of them are not working today."
Ramtane Lamamra, the AU commissioner for peace and security, condemned the attack "in the strongest possible terms".
"The attacks prove that terrorists can hit anywhere, including Africa," he said.
Lamamra said that the body's annual meeting of heads of state would go ahead in Kampala next week.