Your certificate must be identified at least using his Common Name (CN), or at best using his Distinguished Name (DN).
The Distinguished Name is based upon X.500 directory service standard, and is actually a hierarchical set of property/value couples allowing a unique identification of a given certificate, independently of technical certificate properties (key, usages, serial number, validity period, issuer, etc.).
For instance, the Distinguished Name from the current Stack Exchange website is
C = US, ST = CA, L = San Francisco, O = "CloudFlare, Inc.", CN = *.stackexchange.com, which can be decomposed the following way to clearly see the hierarchical relation between each properties (it goes from the more general to the more precise):
C = US // Country = United States
ST = CA // State = California
L = San Francisco // Locality = San Francisco
O = "CloudFlare, Inc." // Organization = CloudFlare, Inc
CN = *.stackexchange.com // Common Name = matches the URL hostname
The details of the attributes composing your certificate will heavily depend on your application and needs (some attributes may even appear several times with different values, the most common example being the
OU (Organization Unit) attribute).
Do not hesitate to check the other question given in the beginning of this answer to get more information on the subject.