The answer seems to be grass-fed cows produce both meat and dairy with more omega-3 and less omega-6, but bio-butter is no guarantee you're getting grass-fed cows.
I remembered Michael Pollan's The Omnivore's Dilemma making a similar claim regarding meat on pages 75 and 267. I followed its citation chain through http://www.eatwild.com to an actual paper, Rule, D. C., K. S. Brought on, S. M. Shellito, and G. Maiorano. Comparison of Muscle Fatty Acid Profiles and Cholesterol Concentrations of Bison, Beef Cattle, Elk, and Chicken. J Anim Sci 80, no. 5 (2002): 1202-11 which analyzes composition of the meat of these animals:
The objective of this study was to compare fatty acid weight percentages and cholesterol concentrations of longissimus dorsi (LD), semitendinosus (ST), and supraspinatus (SS) muscles (n = 10 for each) of range bison (31 mo of age), feedlot-finished bison (18 mo of age), range beef cows (4 to 7 yr of age), feedlot steers (18 mo of age), free-ranging cow elk (3 to 5 yr of age), and chicken breast...
Range-fed animals had higher (P < 0.01) n-3 fatty acids than feedlot-fed animals or chicken breast. Chicken breast n-6 fatty acids were greater (P < 0.01) than for muscles from bison, beef, or elk. Elk had higher (P < 0.01) n-6 fatty acids than bison or beef cattle; however, range-fed animals had higher (P < 0.01) n-6 fatty acids than feedlot-fed animals in [semitendinosus muscles].
So this is a good link between diet and meat composition, and from what I can see there have been several studies confirming the results. Can we find a similar link for dairy products? I dug some more, and according to Couvreur S, Hurtaud C, Lopez C, Delaby L, Peyraud JL. The linear relationship between the proportion of fresh grass in the cow diet, milk fatty acid composition, and butter properties. J Dairy Sci. 2006 Jun;89(6):1956-69, yes:
Milk yield linearly increased with the proportion of fresh grass in the diet (+0.21 kg/d per 10% of grass). Fat yield remained unchanged. Thus, by effect of dilution, increasing the proportion of fresh grass in the diet induced a linear decrease in fat content. Milk fat globule size decreased by 0.29 mum when the proportion of grass reached 30% in the diet. Increasing the proportion of fresh grass in the diet induced a linear increase in unsaturated fatty acids percentages at the expense of saturated fatty acids...
Between G0 and G100 [no grass / all corn and all grass / no corn], polyunsaturated fatty acid (PUFA) percentages increased linearly because of the increase in C18:2 n-6 trans trans from 0.25 to 0.61%, C18:2 cis-9,trans-11 from 0.48 to 1.65%, and C18:3n-3 from 0.22 to 0.70%. On the other hand, C18:2 n-6 cis cis tended to decrease linearly. Very long
chain PUFA (C20 and C22) were not modified by treatments.
Germany's Bio-Siegel requirements say nothing about whether you must feed your cows grass. It bans genetically modified food, irradiation, synthetic fertilizers, and requires certain sustainable practices. But as long as you're growing bio-corn on your farm, you can feed it to your cows and still get bio-butter.
A generous interpretation of Greenpeace's results would be that bio-butter manufacturers do some other processing / selection of their products beyond the Bio-Siegel requirements (such as sourcing primarily from grass-fed cows) to increase omega-3 acids, and that's why they got the results they did. More likely their results are just not statistically significant.