None of these sounds are identical.
The French [œ] in ‘sœur’ is rather similar to the German [ø] in ‘Goethe’. The latter is more close than the former, that is the tongue is located a bit higher in the mouth. [œ] is an open-mid vowel while [ø] is a close-mid one. Both of them are rounded and front. (BTW the French [œ] in ‘sœur’ is almost identical to the German [œ] in ‘Hölle’ and the German [ø] in ‘Goethe’ is almost identical to the French [ø] in ‘peu’).

The articulation of the Russian [ы] (or [ɨ]) sound differs from them in height, backness, and roundness. [ы] is almost close, central, and unrounded, i.e. the tongue is a bit higher and moved more back and the lips are relaxed.