The problem is that I don't have permissions of owner to drill through tiles. Can you advise me, is safe to glue this kind of cabinet to tiles or not? And if is safe, which glue would you use?
In a word no. First issue is you are renting. The owner has specified no holes in the tile. There is no guarantee the adhesive, if it held won't damage the tile when the cabinet is removed. I would suggest getting some sort of a base cabinet, perhaps a bookcase size piece and mount the mirror cabinet to the top .
I would get some foam tape labelled "permanent" and use that. The stuff holds impressively well, and it'll do the job if installed properly. I've seen hundreds of towel and soap dispensers in commercial buildings installed this way for years.
When you're done with the cabinet you'll be able to slide a metal blade behind the cabinet and slice the foam, then scrape away the residue (maybe with the help of an adhesive solvent).
- Clean the tile and cabinet back with alcohol.
- Apply 2" strips of foam tape to every tile (or major portion thereof) that falls behind the cabinet. Orient them vertically. Don't put tape within about an inch of the cabinet sides (where it might be visible), and don't put tape over the grout joints (where it'll be difficult to remove). Keep it on the flat portion of the tile.
- Mark a plumb line where one side of the cabinet will be located. This could be done on masking tape so you can strike a complete line. I say a plumb line rather than a level line because that side of the cabinet is longer, so you'll have greater accuracy.
- With a helper, position the cabinet over the tape and aligned with the plumb line. Do not make contact with the tape yet.
- Slowly ease the side of the cabinet near the mark back to the tile. Make any final position adjustments, then swing the entire cabinet back, making contact with the tape.
- Firmly press the cabinet back into each piece of tape to ensure a good bond.
- Wait a few minutes for the adhesive to grab well, then test the bond by tugging on the cabinet.
The most likely failure point will be the cabinet back. If the paint is particularly slippery (like a vinyl coating), or if it's not clean, you may see the tape not holding. It may be wise to test this beforehand. I have no such concerns with the tile.
Be sensible about what you load the cabinet with. 30 pounds of lotions and warm day could prove troublesome.
Occasionally test the bond with a tug.
Instead of drilling through the tiles to hang the cabinet, you could also drill through the grout lines only. Obviously, you will need a small enough drill and screws to not damage the tiles, and depending on how the cabinet mounts, it may not allow you to place screws in the right location. But the grout would be easier to repair than tile (though it might be tough to get a perfect color match).