Your pear cultivar is grafted on a root from some other tree and these suckers are from that. Prune them away.

You had better remove those green wraps from the trunks, too. Keep that sucker growth as soon as you see it. Do not let it grow as big as you've allowed it to this time.

That space looks inadequate for that tree's mature size, by the way.

Indeed as esh_ga states, with the given proximity to the fence, the trunk of that pear will be rubbing/touching the fence in a decade or so. Now is the best time to move it or take it back for a refund/exchange if possible.

Ditto what Huggorm and Rhizo said about clipping the suckers.

I myself am wondering why its a stick? IMO with or without staking you may be setting yourself up for it to snap. When the tree was planted it probably should have been cut down to branches about 2-3 feet away from the graft. This forces the tree to push out branches for proper form.

ALso, yes those are suckers. Ive seen people dig them up and graft onto them but the easiest thing to do is to cut them down.

How far away from the fence is it? From the picture it looks like less then a foot, but its hard to tell because of the angle.

I suspect that is an ornamental pear, not a fruit-bearing one, canadianplant.

It just under 2 ft at the closest point to the fence behind it and over 3 feet away from the side.

It is not supposed to need that much ground space. The neighbors have one that has grown near 20ft tall and the branches do not start getting wide until it is well over the fence line.
Their similar tree is also close to the fence in their patio and not touching the fence even at its very large height.
I am expecting it to look something like the ones shown in the lined web page after several years and I want to prune it so the branches are not even as low as pictured.

Here is a link that might be useful: Mature Pear Tree

I snipped it off last month and now it is back again.
How can it be removed so it doesn't keep growing back?

suckers will continue forever ...

i think there is some VERY EXPENSIVE GOO you can get... but for $100 [just a number for e.g] .. you may as well just go buy a new tree.. and who needs a gallon of such ....

i heard such discussed in regard to suckering of crabapples... that might help your search ....

ken

"Mature Pear Tree"...Mature callery pears sure don't look like that around here. Picture one at least three times as wide, with a couple of major branches broken off on one side, and loaded with really bad smelling fruit. Also, the field behind would be loaded with annoying, thorny callery pear seedlings making mowing with pneumatic tires a challenge. No, I'm not a callery pear fan at all! The ones in the picture, at the link above, are fairly young.

Below is an example of a chemical for reducing suckering. Amazon has it for $20 plus shipping. You might be able to find it locally at a larger nursery or farm supply store.

Here is a link that might be useful: Monterey Sucker Stopper

I cut them again.
I'm thinking about trying a product called Sucker Punch. Has anyone tried that or a similar product?
What is the best method to apply these products?

Here are new pictures after I cut them again today.
Now they are on both trees, but one has a lot more.

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The product I'm familiar with is called Monterey Sucker Stopper.

Think the process is a lost cause because it won't be long before the expanding trunk takes out the fence.

It is not as close to the fence as it may appear in the photos. I can walk between the tree and the fence.
A neighbor with a similar lot has the same type of tree grown to what looks like over 30 ft tall and it is planted in a similar spot and the trunk still has room to spare. The branches of these trees will all be over the fence line when they get taller in a couple years. The trunk fits in that spot fine.