It's not the EU Investment Bank
My guess is the author is a remainer who just can't help but look for every clue that leaving is a disaster. The name of the entity is the European Investment Bank. There's a difference and the relevant part is that the real name does not contain the word Union. From their 'Projects' page:
"European Union Member States receive about 90% of our financing, but we also work with more than 150 other partner countries in support of EU external assistance policy."
There 'Structure' page has more relevant information:
"The EIB funds its operations by borrowing on the capital markets rather than drawing on the EU budget. The Bank enjoys decision-making independence within the EU’s institutional system.
"The EIB’s management and control structures reflect this independence and allow it to take lending decisions solely on the basis of a project’s merits, and tailor borrowing in line with the best opportunities available on the financial markets."
So they borrow money from capital markets. Therefore they must make a return. If lenders have confidence in the shareholders, they will lend at rates that are competitive. So the board of directors has to decide whether having the European country of Great Britain as a member allows them to secure funds at a better rate or not.
My guess is that Britain is still an asset to the bank and as a result there will be no motivation or appetite for a change in shareholders. Plus, do any of the other countries really want to put up funds that could be spent on domestic priorities in order to buy Britain's share? Would Britain?