By no reasonable standard did the Obama administration have a policy of separating minors.
To make sure we set the scene, the current (2019) Trump administration has a policy that when children and parents are apprehended together, the children are removed from their parents and incarcerated separately from them. The parents are offered no choice in this matter. The claim is that the Obama administration policy is in some way similar.
We also have statements from Politifact that the Obama administration did not have such a policy. Politifact is a good source, and has done much more research than we have. Merely mentioning them in the question doesn't invalidate this status.
The only similarity is that Obama administration deported undocumented migrant parents who had documented US-born children, allowing the parents the choice to take with their children with them or leave them behind in the US. This is the same policy that all previous administrations have had, and which they legally cannot prevent.
Consider the following reports:
HuffPost May 13, 2012 - Deported Moms With American Children Separated On Mother’s Day
About 22 percent of all undocumented immigrants deported in the first
half of 2011 were parents of U.S.-born children. As part of the Obama
administration’s record-setting deportation year, between January and
June 2011, 46,486 undocumented mothers and fathers received orders to
leave the country and were forced to decide what to do with their
U.S.-born children. While in some instances, deported parents decide
to bring their children with them, parents thinking it was in their
child’s best interest, left them with relatives, neighbors, or friends
in the United States.
In the cases above, non-citizen parents with US citizen children are to be deported. The parents are offered the choice of having their children leave the country with them, or have the children remain in the country with other relatives (as is the children's right). It is entirely the parents choice which option they take. They can choose to have their children remain with them if they wish.
American Immigration Council April 26, 2010 - The Ones They Leave Behind: Deportation of Lawful Permanent Residents Harm U.S. Citizen Children
In the Child’s Best Interest, a new report by the International Human
Rights Law Clinic and the Warren Institute at UC Berkeley School of
Law, and the Immigration Law Clinic at the UC Davis School of Law
finds that: More than 100,000 children were affected by parental
deportation between 1997 and 2007.
In a similar way, when parents are deported they can choose to take their children with them or not. The choice is entirely the parents, not the administration.
WASHINGTON (Reuters) JUNE 9, 2014
The Obama administration on Monday announced it is designating a third
U.S. military base for emergency housing of children immigrating
illegally into the United States without parents or relatives, as the
cost of caring for these minors escalated.
Senior administration officials, who asked not to be identified, told
reporters that an Army base at Fort Sill, Oklahoma, will initially
hold 600 “unaccompanied minors” and eventually will be able to
accommodate up to 1,200.
In the above case the detention facility was being constructed for unaccompanied children apprehended at the border. We can speculate about how the children came to be separated from their parents, but it was clearly before they were apprehended by US authorities, and there is no indication that US authorities were in any way responsible.
Under all previous administrations there will have been rare occasions when parents have to be taken into federal detention (perhaps because they are wanted for an unrelated federal crime, or are suspected of terrorism) and in those circumstances their children will be removed from them, since the law mandates that. The difference is that the Trump administration has a policy of taking all detained immigrants into federal detention, which requires them to be separated from their children.
To summarize: The Trump administration forcibly separates children from their families whenever they are apprehended, and detains them separately. The Obama administration did not do that. The only separation was when undocumented parents were deported they were offered the choice to take their children with them or leave them behind (if there was someone to care for them). Some parents chose to leave their children behind.