It was Trump who actually went against Assange at the end of the day. The extradition was started under his administration. Obama's administration did decide not to go forward with it.
It sounds like people are arguing about which President to blame for the jailing of journalists when both parties in the US hold the same position on the matter. The details are circumstantial.
According to all records on this, the reason he was not charged by Obama was that his administration was afraid the case would not hold up in court. If they felt differently, I don't see why the extradition request wouldn't have been made, whether he was still held up in the Ecuadorian embassy or not. After all, Ecuador and the US have had an extradition treaty since 1872 and all that the US had to do is pressure Ecuador and wait/influence them till a new government was in place.
It was Trump's administration that according to all information simply felt differently about the case entirely. This is why his Justice Department went forward.
>President Barack Obama’s Justice Department had extensive internal debates about whether to charge Assange amid concerns the case might not hold up in court and would be viewed as an attack on journalism by an administration already taking heat for leak prosecutions.
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>But senior Trump administration officials seemed to make clear early on that they held a different view, dialing up the rhetoric on the anti-secrecy organization shortly after it made damaging disclosures about the CIA’s cyberespionage tools.
AP(1)
I mean if you have sources to the contrary, please share them. But all that was available so far doesn't really help your assumption there. Also, is this really surprising? Trump promised (but thankfully failed) to "open up Libel laws" after all.(2)