Before I dwell about the PSD2 and its possible effects on the banking ecosystem, a word of preface - the online access to your bank account is still not ubiquitous, but in the EU, the majority of banks provides it, and in some countries (like Poland) it is provided 'for free'. The functionality of that system lets you at least to check your account balance and to make a transfer/payment if only you know account number of the recipient party (which are pretty much standardised across EU, except a few individualistic countries such as the UK).
Banks occupy a very privileged position - they are expected to be institutions of public trust, and therefore receive numerous bonuses, such as f.e. subsidies or limitation of competition. Of course, not all of those bonuses are blatant. The limitation of competition is disguised in regulations - it is not only difficult to start a new bank due to high capital requirements, but even a new product, if regulators are not happy with it.
Of course, it would be tempting to automate this, and there are multiple startups that try to do so, but... banks make it as complicated as possible. Firstly, virtually no bank offers secure API for accessing your financial information. If you want to share it, you have to share your login and password, which banks are advocating against, and indicate it is a security issue. Such trust erosion and technical difficulties (web scrapping is unstable) hamper the FinTech growth, and protect banks.
(1) As soon as this barrier to entry is removed, services providing aggregated financial information (from multiple banks) and helping you manage them will flourish.
(2) Who has customers, has power. Users adopting 3rd party apps will shift away from banks, reducing them to quite expensive datacenters, at least in the area of personal accounts.