The username property for a user is unique, as far as the LMS is concerned, only within the bounds of the org. If the client site is doing user auth with the LMS itself, the username cannot at all be guaranteed to be unique in any space outside the LMS. If the client site is doing user auth by having the LMS use some other Identity Provider for authentication (for example, a CAS or SHibboleth integration, or similar), then it is possible that a single username might have wider uniqueness (within the federation of services that defer user auth to this IDP, for example), but the LMS really has no way of knowing or assuring this. Therefore LMS 1 and LMS 2 have no real way of guaranteeing that username LindsaySmith is the same person identity, or not, as on the "other" LMS.
If both the LMSes defer to an off-board IDP to do user auth, then you might make the understood assumption, by convention, that LIndsaySmith's user identity in LMS 1 and LMS 2 both point to the same person, but that would only be a business rule established by convention.