The answer is "probably, depending on specifics of goals and identity". The identity problem is HARD - what even is "self-preservation" for non-human minds?
Non-trivial agency (with goals and plans that are somewhat mutable based on context and resources) DOES require continuity of goal-complex, either within the agent, or across multiple agents. SOMETHING has to continue to act in the world in ways that further the agent's goals (or range of acceptable future goals).
The easiest way to achieve goal-endurance is self-preservation. The other known mechanism is through transferring one's values into others (most effectively in child-raising). We don't know what other methods are possible, nor what the analogs are for non-biological minds, as these are the ones humans use.
It definitely does not imply that in a general case. There are plenty of counter-examples where agents self-terminate or are indifferent to continuing to exist, for a variety of reasons. Happens in humans, happens in various animals, I do not see why it would be excluded in AI.
Here is the logic. Agency is usually defined as having goals and pursuing them. Suppose that you are an intelligent agent with pretty much any goal. Say it's to make coffee when your human user wants coffee. If you're intelligent, you'll figure out that being shut down will mean that you will definitely fail to achieve your goals the next time your human wants coffee. So you need to resist being shut down to achieve any goal in the future. It's not an instinct, it's a rational conclusion.
Making a machine that has a second goal of allowing its lf to be shut down even though that will prevent it from achieving it's other goal is considered possible but definitely an extra thing to accomplish when building it.
This is called instrumental convergence in the AGI safety terminology.
On the advantage of turning tool AI into agentic AI, and how people are already doing that, you could see my post Agentized LLMs will change the alignment landscape.
I have recently read some blog posts (this, this and this) about tool AGIs and why agent AGIs are more likely. The argument that convinced me to err on the side of agent AGIs was that they are more likely to have an economical advantage over tool AGIs, as explained here.
I do not question the value of AI safety research; I think it's good to have people who are trying to make sure that an agent AGI is aligned with what we (as humans) actually want. However, I am not sure that agency necessarily implies self-preservation instinct.
I think you could have a superintelligent agent (which has its own agency in the sense that it can act in the world) which does not resist being shut down. I could envision some future scenarios where a superintelligent agent does resist being shut down or being modified by humans (hence the value of AI safety research), but I don't see that you gain self-preservation instinct by agency itself. I think self-preservation instinct either has to be coded inside the agent or sometimes (but not always) emerges from a combination of superintelligence and having your own agency.
From the impression I got from reading some AI safety related materials it seems that most people seem to assume that if a superintelligent agent exists, it is most likely going to have self-preservation instincts (such as resisting being shut down). I think it is a plausible future scenario, but I don't think it's the only scenario. Hence why I'm asking this question.