Rather than being named after the 2nd Baroness Ashburton (as suggested in the file page) who died in 1857 over 30 years before this rose was introduced, I'm wondering if it is more likely named after her sister-in-law the Dowager Lady Ashburton who died 15 December 1882 in Paris. She was born Hortense Eugenie Claire Maret de Bassano and married Francis Baring on 29 December 1832 who became the 3rd Baron Ashburton in 1864 and who died in 1868.

Being French-born and also living her last years in Paris too, from what I can tell, is also consistent with Charles Verdier of Paris being the introducer of this rose. There's still over 7 years between her death and its introduction, but there is that curious reference in 'The Rose Garden' of 1903 which says it was "introduced between 1874 and 1902" so perhaps it had been named to honour her not long after her death but not introduced until 1890.
I can't be 100% certain but I can't see a more likely possibility. The 4th Baroness Ashburton, her daughter in law died in 1930.

Parentage is 'Adrienne Christophle', according to the journal of the Société Nationale d'Horticulture de France, 1891, p. 426.