Owners - yes (subject to lease). Tenants on a lease - yes. People living at the premises who aren't named tenants - no.
'Exclusive possession' is formed on the right to occupy, not who does actually occupy - as afforded by the law. It is most often used in terms of a lease, so I'll frame the explanation that way (and is often used to contract a 'lease' versus a 'licence').
When you lease a property the owner grants a contractual right to the tenant to occupy the premises subject to (a) the terms of the lease and (b) the law. That contract creates and gives the right to 'exclusive possession'; along the lines of 'as long as you comply with the terms of the lease, you will have exclusive possession of the premises for the period of x months'. Exclusive possession in basic terms gives you the right to exclude other people from the property, unless they have an overriding right to be there or the consent of the holder of exclusive possession. This includes the owner of the property - unless the owner is properly exercising their rights under the lease and the law (e.g. allowable inspections at reasonable times) then the tenant can legally require them to leave.
Like all contracts this brings in the concept of 'certainty' (specifically, certainty of parties). On one side is the owner - being the registered proprietor of the land - which gives us certainty as to who they are. On the other hand, there are the named tenants in the lease - giving us certainty as to who they are. While other people may be entitled to reside there, with or without the express consent of the owner, those 'add on' people are not parties to the contract (lease): they're not named as tenants in the lease. Subject to an intervening mechanism (e.g. the operation of a law giving them specific rights, or a court ordering the contract be amended to include them as a party) those 'add on' people:
- Are not liable for breaches of the lease;
- Are not liable to ensure payment of rent;
- Cannot enforce the lease against the owner; and, the relevant one here
- Do not get the benefit of exclusive possession of the premises (unless there is a court decision I'm not aware of that gives 'lodgers' and 'boarders' the rights of an exclusive possessor - which is entirely possible).