Estimating the size of Product Backlog items can be considered as a
Product Backlog refinement activity?
If your team estimates, then yes, estimation or sizing Product Backlog Items would be a Product Backlog Refinement activity. Scrum doesn't mandate any particular attributes for Product Backlog Items nor the act of estimation. Refinement moves a Product Backlog Item closer to being ready for selection at Sprint Planning, and the only mandated criteria for being ready is that the Product Backlog Item can be done within one Sprint.
Who is accountable for Product Backlog refinement ?
Everyone on the Scrum Team. However, the primary actors in refinement are the Product Owner and the Developers, and these people have the most interest in making sure that refinement happens and happens well.
The Product Owner is accountable for creating and communicating Product Backlog Items, and refinement is the opportunity for everyone on the team to come to a shared understanding of what the item means. Developers are accountable for planning the Sprint, and refinement activities make sure that the team has a sufficient understanding of the work in order to plan it and then carry it out. The Scrum Master helps the team to focus on create high-value increments, and well-refined Product Backlog Items help the stakeholders to understand what the team is working on, the Product Owner to be able to effectively order the Product Backlog, and the Developers to plan and execute a Sprint.