Or is the background blur something different?

Yes.

When considering blur, there are two numbers of interest.

One is at what distance from the focus distance the details are "acceptably sharp". This is somewhat arbitrarily defined with the circle of confusion. Of course, if you zoom in to an image taken with high-megapixel camera and sharp lens, you will see that nothing is sharp except the details exactly at the focus distance. This is called depth of field. Note that part of the depth of field is in front of the focus distance, and part of it is in back of the focus distance, and the distribution of these may not be exactly 50%/50%.

Another is the blur of the background. In this case, the background is assumed to be at infinity. This is measured with the blur disc size, measured either in millimeters or as a percentage of the sensor size. Using percentage of the sensor size is better, since it allows you to compare background blur on APS-C and full frame.

An f/2.8 lens with same framing indeed always gives the same depth of field (except in cases past the hyperfocal distance, or maybe near the hyperfocal distance). But with longer focal lengths, there is more background blur. Background blur is essentially given by the aperture opening. For example, 200mm f/2.8 is 200 mm / 2.8 = 71.429 mm, but 70mm f/2.8 is 70 mm / 2.8 = 25 mm. So more aperture opening at 200mm, and more background blur, even if the depth of field stays the same with equal framing.