Is it safe to drill in this sliding door frame?
If there are wooden 2 x 4s or 2 x 6s behind the metal door frame, then you will not need anchors. I would use 1-1/2" to 1-3/4" wood screws. With a small enough drill for a pilot hole only for the screws you are going to use, drill a hole in the metal and continue to a depth of 2" from the surface of the metal. If there is wood behind, you should get wood shavings out the hole or clinging to the drill bit.

If drilling this pilot hole reveals substantial wood members behind the metal, enlarge the hole in the metal with a larger bit to allow free passage of the screws through the metal. You do not want the screw threads to engage both the metal and the wood (if there is an air gap between the metal and wood behind) because this could deform the metal.

When tightening down the screw to secure the bracket, do not tighten past snug (do NOT over-tighten) because this could deform the metal, if there is an air gap between the metal and a wood member behind.