Yes, the hub is identical
Same 5x120 PCD and same 72.5 mm centre bore: wheels from BMW 1 Serie F21 bolt on and centre on BMW 5 Serie E61 without rings. You still need to check the offset and tyres, detailed below.
≈: width estimated from the original tyre size.
Wheel by wheel: offset and tyres
Each original wheel from BMW 1 Serie F21 is compared with the one from BMW 5 Serie E61 of the same diameter: how much closer the wheel gets to the shock absorber or how far it sticks out of the wheel arch, and whether the tyre keeps the same overall diameter.
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6.5Jx16 ET33 · 195/55R16 — Check on the car
Compared with the original ≈7Jx16 ET20 on BMW 5 Serie E61: shock absorber side +6.7 mm, wheel arch side −19.4 mm, tyre diameter −5 % : fit it with the original tyre size of BMW 5 Serie E61 (225/55R16), not that of BMW 1 Serie F21.
Comparison for guidance only (estimated width). Check in detail
Before fitting the wheels
An identical hub is not enough: offset, width and diameter change the position of the wheel, and larger calipers can touch a smaller wheel. Original wheels often use bolts or nuts with a specific seat: keep the fixings intended for the wheel, with the thread of BMW 5 Serie E61 (M12x1.50), tightened to the torque shown on its vehicle page. Finally, check that the size is among the approved fitments for your vehicle. This comparison is for guidance only: have the fitting approved by a professional.
And the other way round: wheels from BMW 5 Serie E61 on BMW 1 Serie F21?