Going to add on and critique some of the analysis Inferno made.
Options per Situation
Approach
Ike: The most common and best idea is to space fair. It has a lot of range and some good damage output. Unfortunately, fair is very slow and can be shielded on reaction.
Mario: Use fireballs and/or space bair until you get in. Ike can't swat them like some other characters because he doesn't have a move fast enough to swat a fireball and not be put in a place to be easily punished. Ike should shield them.
Ike: This is actually false. The most common approach is actually N-air since it true-combos into Jab or B-air if it's a blind N-air. And of course from jab you can cancel into grabs for more damage output. Ike can also Running Shield to approach from the ground, so he can easily use his jabs.
Mario: Applying vertical spacing with D-air to avoid getting jabbed works, and also using D-air to harass Ike's shield and defensive style is effective.
Defense
Ike: A move that hits his shield is able to either be grabbed (if its range isn't bad. I know its range) or a jab lock. His jab is really good and stops a fair amount of our ground game. If Ike reads Mario decently, we'll get punished by some jab locks.
Mario: Camp him with fireballs. It will effectively force an approach. Punish his fair when he does approach. If you are close enough, hit him before he hits you. If you aren't just shield it on reaction. FLUDD will also mess up his spacing.
Ike: Few problems here. Mario cannot be jab locked legitimately unless you don't do anything. Mario can easily just Up-B out of it, so Jab locks are out of the question. If you use vertical spacing with B-air (90 degree angle), then Ike cannot shield grab it.
Mario: Since N-air wasn't talked about I'll say it here. An Ike that spaces N-air cannot be punished on block. Simply said, hitting with N-air as soon as you hit the ground allows Ike to immediately go into jab, but he can also shield to avoid punishment. To punish a spaced N-air you'll have to use U-tilt or (DAC) U-Smash.
Edgeguard
Ike: I think fair can be used, but not fully sure. Dair is a very strong spike.
Mario: Hit him with quick aerials and/or fireballs. It's very effective against Ike
Ike: F-air can indeed be used. B-air edgeguard while easily telegraphed works as well. D-air is obvious, and one should always look out for for Aether-spikes if Mario manages to recover high. Eruption edgeguard can be surprising at times as well, especially if you somehow don't ledge-snap correctly.
Mario: Fireball > F-air is pretty awesome for this match-up since Ike pretty much has no defense against it besides air dodge.
Recovery
Ike: Pretty bad. Ike is practically forced to recover high because Mario can easily gimp him if he recovers from low or mid level.
Mario: It's decent. Sort of one-way if you know what I mean, but he has mix ups that he can do to help him.
Mario's recovery is very safe. Ike can't really do much against it if Mario is spamming fireballs, cape stalling to avoid eruptions and aetherspikes, and can practically beat all of his aerials with Up-B.
Margin of Error
Degree of Punishment
Ike: Jab lock. It can punish some approaches and rack some good damage on us. His smashes can be used to punish air dodges, side steps, etc. and kill at very low percents
Mario: Can hit him with fast pokes before he can do much. This will end up turning into combos, juggles, and strings. Most of his moves are slow enough to punish like this. Stutter-stepped Fsmash can be used to punish whiffed moves since it outranges most of his moves
Ike: Already addressed Jab lock from above, but Jab is pretty much Ike's best punish move.
Mario: Pretty much agreed.
Adding on a sub-category on this part:
Capitalization
Capitalization is basically addressing what both characters can do when either character is wide open for attack. Somewhat goes into the mindgame department though doesn't fully rely on it. I'll fill this out myself to give a decent idea.
Ike: One Word, F-Smash. Ike can use F-Smash as a psyche tactic against Mario, particularly when he's approaching him to attack. One mis-spaced aerial on Mario's part can mean F-Smash from Ike if he prepares for it. Predicting air dodges from Mario can also make him eat charged U-Smashes as well, which can kill quite early.
Mario: Not as strong as Ike's in the strength department, but far better as far as damage output goes. Like it or not, Ike's stuff is susceptible to being powershielded and it leaves him wide open. This allows Mario to let him do pretty much anything to Ike that doesn't have a high start-up (which is pretty much only F-Smash and F-air). Using any laggy attack can easily be punished with F-Smash for kills, and Ike pretty much has to go out of his way to use laggy attacks to kill Mario, if it isn't B-air or F-tilt. F-tilt in particular can still be punished with the Breakdance Combo (Jab-Jab-DSmash) for an easy 21% damage.
Character Attributes
Strengths
Ike: Very strong, sorta heavy around upper mid-weight, long range/disjoints
Mario: Very fast, little lag, has a projectile, very good edgeguarding and gimping
Ike: High damage output. Strong defensive game.
Mario: High damage output due to his combo ability.
Weaknesses
Ike: Very slow and laggy moves, gimpable recovery
Mario: Small range, significantly less killpower than Ike
Ike: Trouble landing KO moves despite the high damage output. Poor edgeguarding. Very readable methods.
Mario: Hard time dealing with Ike when he has the momentum (as in, when you're continuously getting hit because you can't get into range).
Covering Weaknesses
Ike: I don't know fully.
Mario: DO NOT SPOTDODGE, AIRDODGE, OR ROLL!! These three things are what primarily get punished by Ike's smashes. Don't do these and you'll get hit by much fewer smashes.
Ike: Use only quick moves the majority of the time, which is N-air, Jab, Grab, B-air, and F-tilt. Abuse your range for the entire match. Wait for an opening to get a kill instead of risking a laggy attack.
Mario: I wholeheartedly agree.
One thing to note is to make sure you effectively camp to demand an approach out of Ike if you can't get inside.
Kill Potential (Gimping would also go here)
Damage Power and KO Power
Ike: Very strong smashes. Will kill at low percents. Spike will kill at low percents also. Other moves have pretty good kill power.
Mario: Mediocre. Probably kill him at 110% minimum from Fsmash. Usmash at a bit higher.
Ike: N-air > Jab > Grab gives good damage. Pretty much all his aerials have high damage per hit.
Mario: Damage output is
far superior. Can juggle Ike immensely and has more legit true combos.
Gimp Ability
Ike: His spike is all I can think of right now and that won't hit much.
Mario: Very good. FLUDD destroys his recoveries by pushing his aether back making him unable to grab the ledge or by pushing him back during quick draw freefall state
Ike: D-air and Aetherspike pretty much.
Mario: Cape on Aether or, hell, cape on him in general off-stage equals an easy gimp. Fireballs make him extremely vulnerable for gimps as well.
Agreed. Hoping for Ike players to refute this.
Questions for Ike players:
2. How often will Ike get gimped?
If Ike recovers high then he shouldn't be getting incredibly gimped, but Mario's F-throw and D-Smash makes recovering high not guaranteed. I'd say at least once per match.