Infinite time period vs zero time period

Even if the length of the pendulum is infinite, the velocity of the bob will be always finite.

As the distance to cover in one oscillation is infinite, the period will be infinite too.

I see no inconsistency.

I think you're getting too caught up on the word infinite here.

What is actually happening with the pendulum of infinite length is that it is effectively in zero gravity, or if you prefer it is effectively on a frozen pond. There is an upwards force that counteracts the gravitational force, and it remains straight upwards as far as the eye can see, never tilting into the plane to provide a restoring force back to the center.

The “infinite amount of time” refers to the time that the restoring force would bring the bob back to the center; the fact that it is infinite means that the bob will not return to the center. If you give it a good kick, it will instead set off in uniform motion in a straight line along the ice. If you pick it up from one place on the ice then set it down very carefully in another place along the ice, then it will remain stationary in the new place.

Contrast with the idea of “zero period” where there is an infinitely sharp restoring force holding the bob in place at one particular position. You kick the bob, it doesn't move, until you kick it so hard that the string snaps. You try to place the bob elsewhere and you break the pendulum because the pendulum string can't go that far.

This does not create any mathematical inconsistencies. Mainly because mathematicians are generally pretty smart about their definitions.

A function $f(x)$ is called periodic if there exists some non-zero $P$ for which $f(x)=f(x+P)$. $P$ is called a “period” of $f$ and is generally not unique since in many cases $nP$ is also a period for any integer $n$. If there is a least positive $P$ for which the above holds then that unique number is called the “fundamental period”.

So, there is no mathematical inconsistency because:

1. periods are by definition non-zero
2. periods are not unique (a function can have many periods)
3. neither the stationary nor the infinite pendulum have a fundamental period

I think the simplest answer is simply that infinite period means that it will take an infinite amount of time to return to the starting position.

So if the length is infinite the pendulum will move veeeeery slowly and at some point it will be back to where it started. But that point, as correctly indicate by the formula for the period, will be infinite, not 0, because - even if by just a tiny tiny little bit, the pendulum will have moved so it's not in its initial position at any finite time $t>0$.

This is surely right if $l$ is looong but finite. In the purely infinite case, math must be handled with care anyway!