In recent reports, the BBC has used this superlative in relation to the City of Birmingham multiple times:
The largest local authority in Europe is to halt all spending other than services it must provide by law such as social care, waste collection, and protecting the vulnerable.
Wikipedia grants it second place after Kent:
Birmingham City Council is the second largest local authority in Europe (after Kent County Council) in terms of the population it covers.
Now Birmingham is a large city, no doubt. but under the European cities, 1.14 million inhabitants place it not even near the top of the list (Place 27, in fact). That seems to imply that the term "local authority" is meant to be used in such a way as to discount the city governments of all other, larger cities. Somehow, they all seem not to fulfill some criterion of a supposedly unitary (?) governance.
As I understand it, the administrative subdivisions of Birmingham as a metropolitan borough, the 10 Constituency committees have a limited ability to decide on some services and budgets only affecting their area. If they do not take precedence as "local authority", in other cities their divisions seem to do so. Are they so different?
Take a look at some examples:
Istanbul (15.9 million) might be discounted here as it is not completely situated inside Europe.
Moscow (13 million) is a "federal subject" of the Russian Federation, but also a city governed by a mayor and a city duma. Its subdivision seems to be quite complex with 12 "okrugs", 125 districts and 21 settlements, but the 146 municipal formations seem to be tasked with "managing local affairs through local self-government", so they might be fit to be called independent local authorities.
(My own home town) Berlin (3.8 million) has also a comparable double role as federal state and city. But its constitution it is also a single municipality (Einheitsgemeinde) and its 12 "Bezirke" are "not territorial corporations of public law, but simple administrative agencies of Berlin's state and city government":
The district board is in charge of most local administrative matters directly relevant to local citizens; however, all of its decisions can at any moment be revoked by the Berlin Senate.
Le circoscrizioni comunali non sono enti locali, in quanto prive di personalità giuridica, ma organi del comune, seppur complessi e dotati di autonomia (e, quindi, di una certa soggettività giuridica, se si tiene questo concetto distinto dalla personalità).
Municipal districts are not local authorities, as they lack legal personality, but organs of the municipality, albeit complex and endowed with autonomy (and, therefore, a certain legal subjectivity, if one keeps this concept distinct from personality).
I could go on with the Juntas Municipales de Distrito of Madrid, the Arrondissements of Paris and many more, but they all seem to have a common theme: While they have some limited autonomy and have a local elected body, their powers of governance are less than those of a "full" municipality.
What makes this so different from Birmingham? Who came up with this title, and what were their criteria?
origimbo, in his answer, pointed out the EUROSTAT Nomenclature of Territorial Units for Statistics, which also defines subdivisions called Local Administrative Units. Its latest validated list is from 2019 and covers EU-28, EFTA and EU candidates (for the last, a few are missing, but none with million cities). Its largest entries, all above 2 million, are
|Paris (see note below)||2,234,105|
Birmingham (pop. 1,137,123) is listed in place 14.
Population numbers for France are missing, but Paris is listed as a single LAU. The number in the table is from an older file. Istanbul, in contrast, is divided into 39 units, the largest being Esenyurt (pop. 891,120, ranked 21).