I've previously posted this question and since then further investigated the issue.
The problem seems to be the latest iOS (13.4.1), which fails to send the client certificate upon authentication requests.
To test this, I've set up a webserver, requiring a client certificate for authentication. The client certificate is installed on the iPhone. If I navigate to the website, iOS states that a client certificate is required and asks to use the installed one. Once I press continue the connection fails as no certificate is presented to the server.
Exactly the same setup on iOS 13.2.2 works as expected. Client certificate is installed, website is accessed, iOS asks about sending the certificate, iOS does so and access works fine.
I've written a bug report to Apple but have yet to hear from them.
Update: iPadOS 13.3.1 also works as expected.
Update: iOS Beta 13.4.5 has the same issue.
Does anyone else experience this issue or is this an isolated case for me? I've even hard-reset the iPhone to check if it was a problem with my setup.