LUSID has a unique identifier for every instrument, called the 'LUSID Unique Instrument Identifier' (aka 'LUID'). An instrument's LUID is generated when the instrument is created, and is unique to your LUSID domain (client).
In addition to LUIDs, instrument must also be associated with at least 1 unique market identifier (e.g. FIGI or a unique ClientInternal ID you have generated) and can be associated with as many non-unique identifiers as you like (e.g. ISIN, SEDOL, CUSIP, Ticker etc). These identifiers can also be used to reference the instrument inside LUSID.