In this case, 'ethnic-conflict' is presumably referring to ethnicity in the same way as it is defined in the Wikipedia article:
An ethnic group, or an ethnicity, is a category of people who identify with each other based on similarities such as common ancestry, language, society, culture or nation.
The group need not be based on race, culture, or religion.
The article on Swiss people states:
The traditional ethnic composition of the territories of modern Switzerland includes the following components:
The German-speaking Swiss (Deutschschweizer), i.e. Alemannic German, historically amalgamated from the Gallo-Roman population and the Alemanni and Burgundii, including subgroups such as the Walser. Closely related German-speaking peoples are the Alsatians, the Swabians and the Vorarlbergians.
The French-speaking Swiss (Romands), traditionally speaking Franco-Provençal dialects, today largely assimilated to the standard French language (Swiss French), amalgamated from the Gallo-Roman population and Burgundians (the historical Upper Burgundy). They are closely related to the French (especially those of Franche-Comté). They are occasionally referred to as Welsch in Swiss German.
The Italian-speaking Swiss (Svizzeri italiani), traditionally speakers of Lombard language (Ticinese variety) today partly assimilated to the standard Italian language, amalgamated from Raetians and Lombards. They are closely related to the Italians (especially Lombards and Piedmontese).
The Romansh, speakers of the Romansh language, settling in parts of the Grisons, historically of Raetic stock.
Now, Personally I don't know whether these groupings are generally agreed, or whether there is some dispute. That is not the point.
The point is that these groupings are accepted as ethnic groups by at least some people using that definition. Furthermore, by that definition conflict between these group is ethnic conflict, and Lord Ashdown's categorisation would seem reasonable.