I am not a lawyer.
A boundary is literally the line where your property meets the other property.
A boundary feature is something that can mark where the boundary is - so a wall, fence, hedge etc.
Our neighbour told us...it’s our boundary but his wall
Which doesn't really make sense. What I suspect he means is that on the deeds it shows you as being responsible for maintaining the wall/fence/hedge etc on that boundary.
However, he is claiming that the wall belongs to him - it may well do.
The neighbour says the fence is ours.
Again, that sounds very odd. I can only imagine - if what he says is true - that at some point in the past the fence was in a state of disrepair and he, or a previous owner of that property, paid for the wall to be built to replace the fence.
But then why just build it part of the way?
it’s clear his wall has been built over the boundary line onto our side.
You need to be careful with plans shown on deeds.
If the boundary on the deeds lines up with some clearly identifiable feature (eg the corner of a building) then you can place greater reliance on it.
On the other hand, if the boundary is shown as going between the middle of two detached houses then the boundary shown on the deeds can often be out by +/- 50cm.
Are you talking about a difference of 50cm or a couple of metres between two detached houses?
Or perhaps this is a semi-detached or terraced house?
This all makes a big difference.
it’s clear his wall has been built over the boundary line onto our side.
May I ask by how much?
If, for example, you're in a semi or terraced house and it's a matter of a few cm then it's really no big deal.
If it's more like 50cm and the wall is clearly on your side and attached to your house rather than his then it's different. If this is the case then he has built a wall on your property and you can just say thank you for gifting you a pile of bricks that you are free to do whatever you want with.
If you're in a detached property it may be more difficult to determine that the wall is actually on your property unless there is some evidence on the ground that there was a fence formerly in a different position from where the wall is now.
he’s making it clear it can’t attach to HIS wall.
If - and I do mean if - it turns out that the wall does belong to him then he is correct, you can't attach anything to it without his permission.
However, it's a simple job to place a post adjacent to his wall without touching it in any way. You can even do it as a DIY job.
There's a woman that I follow on youtube that does a lot of DIY and here is her video of how she replaced her fence that was in disrepair with a much more sturdy one with concrete posts (although she did need her DH to hold the posts upright - so she didn't do it all on her own):-
What rights do we have if this is the case?
Again, I am not a lawyer. If it is very clear indeed that the wall is built way over the boundary onto your property then he has gifted you a nice pile of bricks that you can do what you wish with.
If, given what it says on the deeds about you maintaining that wall/fence is correct, then unless he can show to the contrary that either he or a previous owner of his property built that wall then the presumption is that it is your wall.
In that case you can do what you like with it and attach anything you want to.
If he can prove that he or a previous owner of the property built the wall and it is, more or less, on the boundary then he is correct and you cannot attach anything without his permission.