There are two ways to write dates in Basque according to Euskaltzaindia (The Royal Academy of the Basque Language). For example Urriaren vs. Urriak for October.
Thus, in CLDR data, we should distinguish between Standard vs. Formatted for Month names in Basque.
When it's simply the name the month, without the day or year as in the top part of the calendar use the nominative form (e.g. October is Urria); Thus, the Standard form: https://st.unicode.org/cldr-apps/v#/eu/Gregorian/4800442b9cbdfba4
Ergative case in date formats with the month and day; thus, the formatted form: https://st.unicode.org/cldr-apps/v#/eu/Gregorian/9aa3b64aaf1346b
Using the October (Urriaren vs. Urriak), here are some examples of use cases:
Urriaren: standard names
On the 20th of October we are going to the beach -->Urriaren 20an hondartzara joango gara.
Month names on a calendar
Urriak: Used only in cases when due to the structure of the sentence there is no verb.
Tuesday, October 20. -->*Urriak *20, asteartea.
Next Monday, October 20, we are going to the beach.--> Hurrengo astelehenean, urriak 20, hondartzara joango gara.
List of all of the month names in the ergative case to use in the Wide formattting: https://st.unicode.org/cldr-apps/v#/eu/Gregorian/9aa3b64aaf1346b