What is the top of the storage unit material? If it is osb or particleboard, I would glue it.

I would use wood dowels like the ikea ones- I picked up a "Doweling Accessories Kit" at Harbor Freight that has all the tools needed to do this- drill bit, dowels etc for $3.99. Drill holes in the table top, drill holes in the storage unit (I would drill these close to the corner edges) and put the dowels in the holes- you can add glue if you want to make it more permanent.

Instead of securing it permanently, which can damage both the table top and the storage unit, simply try to prevent it from slipping. Try inserting something made of rubber betweeen the two. You could use the wrist section of a rubber glove - or two- the thin yellow ones sold for household chores: cut it off then cut it flat. That will only cost you pennies.

you could also use L brackets to attach the top to the side (under the desk portion) of the storage unit. It wont be fully attached all the way along but that should help prevent it from moving. Or depending what the storage unit is made of, drill from under the top of the SU, into the table top. Just get screws that are long enough to go through the SU, but not long enough to poke out the top of the Ikea top.

When pros install kitchen counters, they use a bead of silicone. I would try that.

I think both items are constructed like hollow core doors, aren't they? Meaning they have thin wood surfaces with air space in the middle. The only really solid sections are around the edges and the corners where the normal hardware is used. (I might be mistaken about this construction for the storage unit, but I think it's true for the Lack.)

So. You need to make structural connections where the structure exists. I'd consider combining the two table tops into one unit using straight metal braces (flat pieces of metal with holes in both ends.) I'd use at least three of them, two in the corners and one in the middle if it takes the screws. I'd also use carpenter's glue between the touching ends of these pieces, and try to clamp them if at all possible. Then you end up with a pretty solid single counter top. You can attach it with screws, glue, silicon adhesive, whatever...

If the storage unit is solid enough, apply the angle brackets directly to the sides. If not, you might need to attach a layer of plywood to the sides with carpenter's glue and screws, paint it to match, and attach the brackets to that. Remember that the brackets have to support the weight of the tabletops AND anything you place on them, and the shear force can pull them out if not sturdy.

Once you assemble this, I'd not put anything heavy on the ends (like a microwave or whatever) unless you add table legs to support the weight. Keep heavy stuff centered over the storage unit.

Good luck! Sounds like an interesting hack!

@SherryBinnNH is right, the tabletops are only solid in the corners where the legs are screwed in.

However, I would use a different glue other than carpenter's, as it's best on unfinished wood. Silicone or E6000 to glue the two edges together (using clamps), and mending plates where the leg brackets are attached. Position the mending plates where you can still add shelf supports to the solid area.

If this is something that will support a lot of weight, consider using their solid-wood LAGAN countertops instead. For this, you can just screw it to the top of the shelf from underneath. For these, carpenter's glue is appropriate.

Must the tabletop be secured TO the storage unit? If not (if you're only concern is slipping/sliding) place the tapletop(s) face down on the floor, set the storage unit on top & mark all sides. Glue furring strips or scrap pieces of molding along your lines. Allow ample time for curing, flip & assemble. Feasiblility depends on the ultimate purpose of the unit...and whether you have wee climbers under your roof - in which case you should toss this option.

Velcro stick-on strips will joing top and bottom surfaces without damaging them. In builders supplies or hardware stores. One inch squares and larger sizes so you can cut to the size you want. Won't budge. Use pliers and pull to remove them. Goo Gone or lighter fuel will remove glue residue.

I second @Skidou's velcro suggestion. I put my desk together that way (also with the hollow IKEA tops - one corner piece and one rectangular piece off one side), and especially once you set stuff on it, it's so solid, you'll forget it's not actually screwed together.

I don't have suggestions (sorry) but I do wonder what brackets you're using to do this? I want to try something similar in my kitchen to give me some more counterspace..

Another vote for @skidou's velcro suggestion! Look for the bright yellow 3M Command strips (which aren't actually Velcro). They come apart when you pull EXACTLY UP AND DOWN, but not in any other direction, so it's sturdy.

But the easiest thing is probably what @skidou said about friction mats. Just get some rubberized shelf liner from the dollar store, and you're all set.