Assuming you use windows.
The PIN is cached to avoid having to re-enter it. The certificate is on the smart card and cannot be cached as the pin is used to unlock it.
I don't think Chrome can cache the pin as it's handled by the credential providers and not the application itself.
Relevant info from https://docs.microsoft.com/en-us/windows/security/identity-protection/smart-cards/smart-card-architecture#pin-caching:
PIN caching
The PIN cache protects the user from entering a PIN every time the
smart card is unauthenticated. After a smart card is authenticated, it
will not differentiate among host-side applications—any application
can access private data on the smart card.
To mitigate this, the smart card enters an exclusive state when an
application authenticates to the smart card. However, this means that
other applications cannot communicate with the smart card and will be
blocked. Therefore, such exclusive connections are minimized. The
issue is that a protocol (such as the Kerberos protocol) requires
multiple signing operations. Therefore, the protocol requires
exclusive access to the smart card over an extended period, or it
require multiple authentication operations. This is where the PIN
cache is used to minimize exclusive use of the smart card without
forcing the user to enter a PIN multiple times.
The following example illustrates how this works. In this scenario,
there are two applications: Outlook and Internet Explorer. The
applications use smart cards for different purposes.
The user starts Outlook and tries to send a signed e-mail. The private key is on the smart card.
Outlook prompts the user for the smart card PIN. The user enters the correct PIN.
E-mail data is sent to the smart card for the signature operation. The Outlook client formats the response and sends the e-mail.
The user opens Internet Explorer and tries to access a protected site that requires Transport Layer Security (TLS) authentication for
the client.
Internet Explorer prompts the user for the smart card PIN. The user enters the correct PIN.
The TLS-related private key operation occurs on the smart card, and the user is authenticated and signed in.
The user returns to Outlook to send another signed e-mail. This time, the user is not prompted for a PIN because the PIN is cached
from the previous operation. Similarly, if the user uses Internet
Explorer again for another operation, Internet Explorer will not
prompt the user for a PIN.
The Base CSP internally maintains a per-process cache of the PIN. The
PIN is encrypted and stored in memory. The functions that are used to
secure the PIN are RtlEncryptMemory, RtlDecryptMemory, and
RtlSecureZeroMemory, which will empty buffers that contained the PIN.
