The background blur disc size in millimeters on the sensor can be obtained from the slightly simplified equation
b = f^2 / (x_d * N)
b is the blur disc size in millimeters,
f is the focal length,
x_d is the subject distance and
N is the aperture number. In the equation, it is assumed that background is at infinity (if it's not at infinity, the equation would become slightly different, slightly more complicated).
I can further introduce
crop factor into the equation:
b = f/N * (f*crop)/x_d * 1/crop
Now, I want to express
b as a fraction of sensor diagonal:
relative_b = b/diagonal = (f/N) * ((f*crop)/x_d) * (1/(crop*diagonal))
crop*diagonal is constant. So, the first term,
f/N is the lens specification. The second term,
((f*crop)/x_d) is constant for equal framing because
f*crop is the 35mm equivalent focal length, and it divided by the subject distance is constant for equal framing. The third term too is constant: if you multiply crop factor by sensor diagonal, you get full frame sensor diagonal which is a constant.
So, the relative background blur as a percentage of sensor size stays constant for a given lens, assuming you shoot wide open and use equal framing.
Now let's take a look at depth of field. The following equation can be derived for it:
DoF = 2 * x_d^2 * N * C / f^2
DoF is the depth of field,
x_d is the subject distance,
N is the aperture number,
C is the circle of confusion and
f is the focal length.
I can further introduce
crop into the equation:
DoF = 2 * (x_d/(f*crop))^2 * (N) * (C*crop^2)
C * crop is
C_FF, the circle of confusion for a full frame sensor, so
DoF = 2 * (x_d/(f*crop))^2 * (N) * (C_FF*crop)
In this equation,
2 is constant,
(x_d/(f*crop))^2 is constant given equal framing,
N is dependent on the lens and
(C_FF*crop) is proportional to crop factor.
So, the crop camera has more depth of field although it has exact same background blur.
If the photographer desires both large background blur and deep enough depth of field, it actually is beneficial to use a crop camera, or alternatively to emulate a crop camera on a full frame camera (walk back, and crop the final image to be smaller). It depends on the lens whether any crop camera benefit can be seen compared to cropping the full frame image: a poor lens can be the limiting factor in resolution, giving crop camera no edge, whereas a good lens can benefit from the higher pixel density of a crop camera.
A full frame camera can always emulate a crop camera albeit with lower megapixel count (and without crop lens compatibility if it's DSLR and not mirrorless), whereas a crop camera cannot emulate a full frame camera.
Summary: with a given lens, crop camera gets more depth of field, but background blur is the same for crop and full frame, if the background is at infinity.