The financial institution's liability only becomes effective if XXX has granted it a licence to operate.

I.e, becoming subject to liability on the condition that A has done B;

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Note added at 40 mins (2023-01-06 21:53:52 GMT)
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Baumann Avocats Droit informatique
Le mot fait plus particulièrement partie du vocabulaire fiscal. L'assujettissement est la situation dans laquelle se trouve une personne qu'une Loi ou un règlement administratif contraint à une prestation financière au profit d'une caisse publique. En droit de la Sécurité sociale, l'assujettissement est le rapport obligatoire qui lie au système de Sécurité sociale les personnes recevant la rémunération qui leur est versée

Baumann Avocats IT Law
The word is more particularly part of the tax vocabulary. Liability is the situation in which a person is obliged by a law or administrative regulation to make a financial contribution to a public fund. In social security law, liability is the obligatory relationship between the persons receiving the remuneration paid to them and the social security system.

"assujettissement" of A by B means B gaining authority over A.

You can see it as a parallel with "devenir le sujet du seigneur" / "les sujets du Roi".

Yes, it does sound a bit archaic. No surprise there - legal terms resist strongly any novelty.

Comment Appelle-t-on les sujets du roi

Les sujets sont, en conséquence, privés de tout droit de contrôle envers le roi. Ils sont soumis au roi comme ils doivent l'être à Dieu, dont il est le lieutenant sur terre. L'obéissance au monarque est un devoir sacré. Quel est le féminin de sujet ? Sujette est la forme féminine de l'adjectif sujet.

https://bagagesdedonnees.com/bibliotheque/article/read/70847...

See any parallel with the taxman and the taxpayer? Or with any company being "under the authority" of some supervisory body?

In practical terms:

L’assujettissement de l’établissement financier ne devient effectif que si xxxxxxxxxx lui a octroyé l’autorisation d’exercer.
=
Until the licence to operate has been granted to it by XXXXX, the financial institution is not under the authority of (whatever is the supervisory body)

or if you want to keep the same sentence structure:

the (right of) supervision over the financial institution will not be effective until ...

taxable - falling into tax law

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Note added at 2 hrs (2023-01-07 00:04:17 GMT)
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correcting above explanation : falling under tax laws (not falling into tax law)

French term (edited): assujettissement (à)

to throw another idea into the 'regulatory control' mix.

Besides 'subordination or assent to', alignment seems to work well with the preposition of à

L’assujettissement à xxxx d’un établissement financier suit la procédure suivante : a financial institution's regulatory alignment *to* xxx is to follow the procedure below.

conditions d’assujettissement: conditions of regulatory alignment (compliance control) vs. Bridge - qualifying conditions

One thing is for sure. The term in this context has absolutely nothing to do with liability to tax.

Although "assujettissement" often means liable to taxation or tax liability, it is unclear if taxation is involved.

If it's not about taxes as in another suggestion (i.e. liability) , then this is the meaning I give it
example sentences from Collins Dictionary https://www.collinsdictionary.com/us/dictionary/english/regu...

Par ailleurs, quiconque peut, sur la base d’un mandat, disposer à titre professionnel au nom et pour le compte de clients, de valeurs patrimoniales, est considéré comme gestionnaire de fortune. Est réputé trustee quiconque, à titre professionnel, gère un patrimoine distinct ou en dispose en faveur d’un bénéficiaire ou dans un but déterminé, sur la base de l’acte constitutif d’un trust. Pour ces catégories de professionnels, la loi prévoit l’assujettissement obligatoire à un organisme de surveillance.
https://so-fit.ch/

Subordination is "the state of being subject to the authority of someone, especially within a hierarchy". Le Grand Robert refers to the terms subjugation, dependence, slavery, inferiority, obedience, suborder, guardianship, vassalage, and discipline. It adds that subordination is the "submission to a thing". Submission is defined as "the willingness to submit, to obey".
https://www.lawsen-avocats.com/en/subordination-and-relation...

• The subordinate status vis-à-vis non-subordinate creditors increases the default risk.
https://de.products.erstegroup.com/Retail/en/Products/Bonds/...

Subordinate financialisation involves the subjugation of domestic monetary policies to the imperatives of international capital …
:
All of this appears explicitly as a result of the subordinate status of peripheral states in international capital markets and the subordinate status of the domestic currency.
https://core.ac.uk/download/pdf/19090664.pdf

Bankruptcy courts in the United States have the power and authority to subordinate prior claims on the assets of a bankrupt debtor to the claims of junior claimants based on principles of equity. This is a remedy called "equitable subordination."[2] The basis for subordination is usually the inequitable conduct of the prior claimant with respect to junior claimants.[3] Equitable subordination can be used to subordinate both secured and unsecured claims.
https://en.wikipedia.org/wiki/Subordination_(finance)

(1) A Secured Party may, in a Security Agreement or otherwise, subordinate its Security Interest to any other interest.
(2) An agreement to subordinate is effective according to its terms between the parties and may be enforced by a third party if the third party is the person or one of a class of persons for whose benefit the subordination was intended.
https://qfcra-en.thomsonreuters.com/rulebook/article-40-volu...

Subordinate Financing Agreements (Subordination Agreement and Secondary Financing Rider)
:
5. Should the Subordination Agreement to be used in LIHTC deals?…

Yes, if a transaction includes subordinate debt, the Subordination Agreement should be used, regardless of whether or not the transaction includes LIHTCs. The Subordination Agreement is for use between public body lenders.
https://www.hud.gov/program_offices/general_counsel/mffaqs/s...

Creditor is willing to subordinate (on the terms set forth in this Agreement): (i) all of Borrower’s indebtedness and obligations to Creditor pursuant to the Series 1 and/or Series 2 Senior Subordinated Secured Convertible Notes issued by Borrower on December 30, 2005 to Creditor (the “Subordinated Notes”), whether presently existing or arising in the future (the “Subordinated Debt”) to the Senior Debt (as defined below); and (ii) all of Creditor’s security interests, if any, in the Borrower’s property, to all of Bank’s security interests in the Borrower’s property.
https://www.sec.gov/Archives/edgar/data/357108/0001193125060...

Context translation:
The subordination of the financial institution only becomes effective if xxxxxxxxxx has granted it authorisation to operate.