Why past tense in 'I got this'?

The OED has this usage back to 1849 so it's been around a while. It says that it comes from omitting have and is "colloquial":

b. The pa. pple. [past participle] is also used colloq. with omission of (I) have. Cf. gotcha n., gotta v.

1849 Knickerbocker 34 12 They got no principles. They got no platform to stand onto.
1857 Quinland I. 1 Got an hour to spare—thought I'd just run in and see what you were all about.
1884 ‘M. Twain’ Adventures Huckleberry Finn xxxviii. 325 We got to dig in like all git-out.
1887 M. E. Wilkins Humble Romance 370 What you got there, grandma?
1911 R. D. Saunders Col. Todhunter i. 11 Oh, of course, you got to laugh at me.
1911 J. F. Wilson Land Claimers ix. 118 But I got several plans, and I need ye.
1941 P. F. Webster & D. Ellington (title of song) I got it bad and that ain't good.
1967 L. White Crimshaw Memorandum (1968) v. 93 Gawd knows I got enough problems.

EDIT: I don't have evidence, so I didn't originally include it in my answer, but my suspicion is that:

In US informal registers, got seems to have been re-interpreted as a present-tense verb form just meaning "have, possess". It sure behaves that way. It's homophonous with, but not identical to, the past tense of get. Historically it seems to have been a resultative construction, but it acts like a normal verb now.

The only issue is if so, then the verb's defective in the 3sg: both "he/she/it got" and "he/she/it gots" are highly marked and are just avoided in most dialects. In response to Betty's inquiry, I'm not sure sure if people just say "he/she/it's got" or if we reword to avoid the issue.

Again, I got no evidence; it's just a pet theory for now.

"I got this" (or slightly more grammatically "I've got this") means "I've taken this on," "I know what to do about this and I'll do it," "No-one else needs to do anything."

A [rugby] footballer or cricketer (or baseball player perhaps) might call "I got this" when he's about to catch the ball falling from the sky, to indicate that he's made the decision to deal with it [same tense] and other players shouldn't interfere — which runs the risk of no-one catching it. It's now applied to other situations such as that in How I met your Mother.

I don't see it as being Past Tense at all.

People say "got" all the time in casual spoken English, as in "I got your back," "I got a lot riding on this one," "I'm giving it all that I got" etc. Even the ubiquitous "gtg" online (got to go).

None of these is really in the Past.

The often unrecognizable expression is "Have got," which is also different from the Present Perfect (Have P.P.).

"Have got" is the same as "Have." The added "got" is just there for rhythm and doesn't give any special meaning. It's not the same as the P.P. "gotten" (American) and "got" (British).

So it is not precise to say that we are presuming a Past (done deal) Action when we are still in the Present.

Perhaps one could say we are presuming a Present condition when the task is still in the Future, yet to be accomplished.

But the OP has already mentioned that a meeting of will and act in the Present Tense is understandable.

I also found this from BBC Learning English:

'have' / 'have got' When we are talking about possession, relationships, illnesses and characteristics of people or things we can use either have or have got. The have got forms are more common in an informal style.

Have got has the same meaning as have and both are used as present tenses.

The use of "got" in any such usage where the context is present tense, the "got" intends to emphasize the surety/need for urgency.

Surety as in "Don't worry, I got (confidence that he/she already has a plan to take care of the task, so consider it's already being attended) it"

OR

urgency as in "I got (its so urgent that consider me gone already) to go".

That's pretty much i could think, I hope that answers the question.