I'm curious if this is common practice, and if it seems logical. In tournament mainly. Something I have been doing recently is recording when I get a miracle card(or two :-) in all-in situationd. As well as when the villain does the same . Then when I enter a tournament I look at these stats to help influence my decision as to make an all-in move.
The tracker i use (for cash games) track expected EV and eventual win/loss for all in hands, which i think is fairly common. If i've been getting bad beats, then my EV graph will be much lower than my win/loss graph, and vice versa. This is to give a sense of whether my wins/losses can be attributed to good/bad plays, or good/bad luck. If I am losing due to bad plays, then i make an adjustment, otherwise i hang in there. I can see this making sense to track in tournament situations as well.
However, it seems like youre asking if you can quantify "luck" to influence your future actions. This is purely wrong. Thinking you are running good (luck) and therefore should gamble, or running bad (gambler's fallacy) and therefore are owed a win, are both losing mentalities. Your chance to win in a particular hand is not affected by these things. You should be aiming to play a good solid strategy, making adjustments based on how your opponents play, not based on how the board runs out.
There are software tools that help you track your hand history and review your hands, however, I don't think the way you intend to use them is the most useful.
If I understood you correctly, you want to see how many times either players hits a miracle card on the river (say, in Texas Hold'em, 22 vs 33 on a 223K board). Then you'd like to use that information as an estimation of the probability of certain events happening again, thus informing your decisions.
The thing is: tracking your hands is a very ineffective way to do that! You'll need huge samples and great memory! What I will tell you right now is not an exact method, but it's kind of useful:
A poker deck has 52 cards. The two you hold in your hand are known, so only 50 unknown cards. After the flop you know three more, so we are down to 47. After the turn there will be 46 and so on... However, just to make numbers easier to compute, we'll stick to a 50-ish card deck. Now, every hand has (approximately) a 2% chance of being the next one drawn.
For instance, in the example above, the player holding 33 has a 2%-ish chance of hitting quads in the river, as there is only one card that would save him.
Now imagine we have two clubs in our hand and there are other two on the board. The deck has 13 clubs, so there are 9 clubs remaining in the "unknown cards". Since 9x2=18, we have an 18% chance of hitting our flash. If we were still on the flop and two cards were yet to be drawn, we'd have two chances of hitting the flush, so 36%-ish (this calculation is not exact, but it's a good enough approximation. Our real chance would be more like 33%, since we are counting twice situations where two clubs come)
For more info, look for some guide on outs and odds
There exist two core kind of probabilities: independent and dependent.
An example of a dependent probability is an outcome from a deck of cards, where each card dealt has an effect on the remainder of the deck. When you hold two cards of the same suit, the odds of a card of that exact suit being dealt next are reduced because you've removed the possible combinations of cards of that suit. The outcome is dependent on what has already been dealt, i.e. occurred.
An example of an independent probability is an outcome of a coin-flip. You may flip a coin as often as you want, the results are not going to have any affects on the odds of your next flip being heads or tails.
Each hand you play is in essence one big coin-flip. Once the hand is over everything resets. Nothing that has happened in the past can affect what happens to a newly shuffled deck.
A poker site would never, ever, store what happened to you or your opponents in the past, in order to make you "luckier" or "unluckier" to make up for the past. Firstly it would be cheating and the sites would be closed for doing so, and second of all it would be ridiculously expensive to store these records.
[...] if it seems logical [...] I look at these stats to help influence my decision as to make an all-in move.
It is not logical. Storing these suck-outs is pointless. You should rather work on learning the actual probabilities of the game that are bound within each hand. You're really just gambling, using probabilities that have no meaning whatsoever. How on earth could the deck suddenly change that based on something in the past that the deck has no idea about? It can't.
You can learn actual probabilities that tell you with 99% accuracy how likely an event is to occur in your game. Your chances of hitting a gut-shot on the next street are exactly ~8% (4 outs * 2%). Nothing will ever change that, no bad beat or a miracle card in a past hand.