The passage in question:
His own [country estates] were modest enough and less remarkable for their statuary and pictures than for their landscape gardening and the rarities on display: for example, at Capreae he had collected the huge skeletons of extinct sea and land monsters popularly known as 'giants' bones'; and the weapons of ancient heroes.
For /u/i_post_gibberish as well
No, the Latin doesn't say anything about extinction:
immanium beluarum ferarumque membra praegrandia, quae dicuntur gigantum ossa
which is more directly "very large body parts [membrum is usually like arm, leg...dangly bit...] of great sea monsters and wild animals, which are called 'bones of the giants.'"
My guess is that 'extinct' was inserted by the translators to convey the their view of the significance to modern readers of giants' bones--that is, creatures that were unknown to the Romans because extinct (the idea that large fossilized skeletons represent long-ago, long-gone giants is attested in a lot of place, even Augustine's City of God).
What we know for sure about the bones Augustus had/claimed to have is...well, pretty much what Suetonius tells us. However, there's one more detail in the passage that might help fill in some detail, although it's definitely tentative. Suetonius specifies that this particular collection/proto-museum was on the island of Capri (Capreae, or the Capreis you'll see in the Latin). Fossils of mammoths, rhinos, and cave bears have been found by archaeologists on Capri. So if Augustus was drawing his collection locally, which is not a given but seems at least possible, it might well have included bones from those animals.
Followup question: Is this translation accurate? I was under the impression that the concept of extinction didn't even exist until the 19th century when palaeontology started to be a thing, and was seen as controversial at the time because it supposedly refuted the perfection of God's creation. Of course, Suetonius wouldn't've cared if something contradicted Christianity, but if the translation is accurate it implies that the Romans were familiar with the concept of extinction, which raises the question of why it was such a "new' idea in the 19th century.