First, start with Warlock instead of Wizard since the Warlock thematics (bargains with other beings for power) is better than the Wizard thematics (learned scholars harnessing non-sapient energy).
On thing to keep in mind, when this was written, was Sorcerers / Warlocks didn't have a representation yet. Familiars weren't standard for Wizards either, or at least not until higher levels, so Warlock / Sorcerer would both work as better comparisons, yes.
Instead of Spells a Shi'ar has Gen. He can have One Gen at Level one, and gains an additional Gen at 5th, 11th and 17th level (for a max of 4 Gen at one time).
Having multiple Gen has the potential to be broken, but at the same time I can see it being done (especially since Sha'ir can eventually imprison multiple genies and Al-Qadim was a lower-tier fantasy than default 5th Edition). Presumably the benefits that apply for consorting with a Genie of a certain element would only apply if they consistently pick Gen of the same element, or alternatively they can stack (so while a Sha'ir with a Water and Air Gen would, when dealing with a Genie of Water or Air, get a +2 bonus to diplomacy and whatnot, Fire or Earth -2, but if they picked Water and Water then they would get +4 for Water and -2 for everything else).
I guess I should specify how they're stat-lined in AD&D to aid with conversion: AC5 (AC15 in 3.5+ terms, in AD&D you start at 10 then count down instead of starting at 10 then counting up), Movement Rate of 9 (mid-way between Dwarf and Human, note that in AD&D Dwarves are half as fast as Humans so being 3/4 a human's speed it'd probably be simplest to assume a standard Small creature's speed), "Low" intelligence (5-7 in AD&D terms, not sure where that converts to 5th but probably around 6-9), "a number of hit points equaling half its master's current maximum" (Hit Die too), and "the THAC0 of a monster that's half their master's level in Dice" (so the BatB of a monster half the Sha'ir's level). "Gens inflict 1d6 points of damage, and are of small size."
The four sub-types of Gen given are:
Air Gens (Djinnlings), who fly at MV 12 (so fly at Movement Speed 30), "but have no other djinni-like powers, and cannot become invisible or create objects". They tend towards good and lawful behavior.
Fire gens (Efreetikin), can
produce flame at will and tend towards evil / lawful behavior.
Water gens (Maridans), have a swim speed of MV 12 (thus they gain all the bonuses of an innate swim speed, and swim at 30), and can breathe underwater without penalty. They're often good / chaotic in behavior.
Earth gens (daolanin), inflict twice the base damage with their natural attacks (2d6) and tend towards evil / chaos behavior.
Having a Gen provides a bonus of +2 armor class against elemental attacks of the same type as the Sha'ir's Gen, as well as +2 to Saving Throws and -2 damage per
die of that element (to a minimum of 1), so for an example a Sha'ir with an Efreetikin is targeted by a 6D6 damage fireball. They benefit from +2 to their saving throw (to receive half damage), as well as - regardless of the results - reduce the damage of each die by 2 (meaning that the most damage they could suffer is 24 instead of 36, and if they save the most they can suffer is 12 vs 18). This benefit only applies when the Gen's on the same plane as their master and within 10', but applies to the Gen at all times. Regarding saving throws, the Gen counts as being
twice the level of its master, so it's generally a very foolish idea to do something like target an Efreetikin with something like a fireball (since they'll get +2 to their already high saving throw).
Gen must stay within 100 yards of their master while on the Prime Material Plane. If forced to move beyond such it shifts into the elemental planes (when able) and in 10-60 minutes reappears if their master is on the Prime Material Plane, 1-6 minutes if their master's on an Elemental plane, 1-6 days if their master is on neither an Elemental nor Material plane. Unless impossible they appear within 10' of their master when arriving on the same plane. It tends to appear wherever is safest (such as behind the Sha'ir, or on the far side of some enemies), and if it cannot appear anywhere safe it will refrain from returning to the Sha'ir's plane until it is safe to do so (so you can't, say, throw a Sha'ir with 100% fire resistance into lava and wait for their Gen to immolate itself). If the Gen is more than 10' (two squares) from its Master, but within 100 yards, and someone immediately threatens it, the Gen tends to teleport back to its home plane and return within 10-60 minutes. As said prior, Gen and Sha'ir share a telepathic connection.
Gen death is depicted as a little more harsh than standard Familiar death (no Con' loss, but half the Sha'ir's base HP is supposed to be lost with a saving throw to stop at 1HP if this would render them unconscious / dead). There's also a bunch of loyalty / morale stuff, but that isn't as important in 5th Ed. The gist of it is a Sha'ir
can replace a Gen, but while their firsts tend to be fanatically loyal the last in a long line tend to be untrustworthy and jittery. Gen can be charmed and the like, but even when so they will not directly attack their Sha'ir.
Summoning then binding a Gen is depicted as a 1d20 hour process requiring a magical proficiency check, but if the Sha'ir "communes" with the respective element (is in proximity to a lot of [respective element]) and fasts then success is automatic (so a malnourished and unattended Sha'ir in a damp cell, for example, would be able to summon a Maridan fairly easily).
This effectively gives the Shi'ar four physical attacks per turn, like a high level fighter. Probably less powerful since they are only CR 1/4 creatures, even with the attack and damage bonus but that is to be expected since you can safely stand back and order them to attack for you.
Well, "safely" stand back. Losing your familiars in combat, even if reduced to Wizard Familiar-esque penalties, would still be fairly undesirable. Plus a Gen that's physically attacking is not fetching spells.
These spells originate from the Gen (affects that target Self target the Gen, for example).
I feel like pointing out that, at least in AD&D, it would be the Sha'ir who casts the spell. It's just that the Gen effectively snags the raw elemental force, tells you how to channel it, then throws it through you instead of forcing you to channel from (respective magical plane) yourself.
This takes a variable amount of time, during which the Gen is not available to you. Gen return when you take a Short or Long Rest. Gen can retrieve any spell up to 5th level from the Warlock Spell List. You gain access to higher level spells as you level up, exactly as the Warlock does.
Similarly regarding AD&D Gen, there is no limit to either spell level or Arcane / Divine (you can steal Divine spells, if with the risk of negative consequences that scale in terms of Divine Spell's level). However, for simplicity's sake it may be easier to restrict the Gen to Warlock spells. It
should be pointed out, though, that the Gen can only fetch spells in AD&D that the Sha'ir knows about, so you probably couldn't send one in 5th to find some obscure spell from Eberon while you're in Faerun.
Shi'ar do not gain a Pack Boon. At first level the Shi'ar selects a Patron. The nature of this patron determines the creature type of his Gen (celestial, elemental, fey or fiend), grants an extended spell list (Celestial would grant healing spells, for example) and additional boons at levels 3rd, 6th, 10th and 14th. (I can't bother thinking all these up, but they would be roughly equivalent to the Warlock stuff at the same level.)
Once more "feel like pointing out", Gen are intended to be some sort of elemental demi-genie being versus some undefined magical being. If the Sha'ir is intended to be less Al-Qadim specific this sounds good, but if Al-Qadim in nature keeping it Elemental Genie is probably for the best.