Q:

How Chrome browser know which client certificate to prompt for a site?

I'm setting up certificate authentication for my project using Tomcat. It works ok for command line client such as cURL.
I have many client certificates installed in Chrome browser. Some are using to connect to my site, others are used for different purposes and not relate to my project.
Every time I connect to my site, Chrome presents a list of client certificates for choosing. These are exactly the certificates that I installed and not others. My questions are:

How Chrome knows which client certificates are for a site to present for choosing?
Tomcat stores those client certificates in its trust store. During
SSL hanshake, Tomcat will request for client certificate. Does it
request for some specific certificates that it knows in its trusted
store so that Chrome knows what to show?

A:

The client certificate authentication is ruled in the handshake phase of the SSL/TLS protocol implemented by browsers.

If the server requires a client certificate authentication (it is
optional), send a message to client with the list of the accepted
certificate authorities (CA). Can be void if server accepts any
certificate.  
The client select the certificates installed in client keystore which have been issued by any of these CA's, and present the list to user. In case of Chrome, the browser selects the certificates installed by user from the Operative System Key Store. 
User choose a certificate, and the client performs a signature with the private key of the certificate over a known data interchanged during handshake.

Only certificates with private key can be selected during step 2. This is the reason by with the browser does not select the certificates of trusted CA's installed in your device. You do not own the private key

