A diplomat famously pointed out that countries do not have friends, they have shared interests. Similarly, organisations do not have feelings; they have contracts. People have feelings.
The point of the above statement is that, if we take it on board, it helps us to avoid becoming emotionally attached to an organisation we work for and the consequent distress we experience when it no longer requires our services – it is simply exercising its contractual obligations to you and to whoever governs/owns it.
I am not into loyalty when it comes to organisations because an organisation, by definition, cannot reciprocate it – it does not have feelings. The people in an organisation may miss you but the organisation most certainly will not (unless you have something it needs). This does not mean that we should not care or not keep our word – we enter into commitments (contracts) with organisations and make promises and develop loyalties to individuals (internally and externally).
When it comes to the crunch, such as staff cuts and restructures, organisations look to contracts and requirements not feelings.
In terms of looking after your personal well-being, it may be wise to not confuse your personal and contractual obligations because to do so can lead to quite a bit of anguish and stress over ‘unacknowledged loyalty’ – however that may be construed.