Yes, they can, I'm afraid. Its actually illegal to attach a plant, or a support for a plant, or anything else, to a neighbour's fence,because the fence is their property, not yours. If you want a climbing plant, and you need a support, you're obliged to erect your own posts and trellis on your side, not attached to the neighbour's fence. In practice, most people are unaware of the law in this matter, and do what you've done for the last few years. Clearly your neighbours have either just learned about the law, or always knew, but never objected when it was the old fence, or never realised you'd attached something to their fence.

Drive some wooden posts into the ground - on your side and attach your trellis to the posts, problem solved.