A very rough calculation (disregarding minor things like leap years):

(1) Let us assume on average a five year time period during which two grandchildren to the same woman are born.
(2) Let us assume when they say 'the same minute' in the newspapers they don't mean it verbatim, and that the timing of births is often not done exactly: let's say 10 minutes real time could still lead to reports of 'in the same minute'.
(3) Let's be very generous and assume only one living grandma per child (if not there would be two chances for to coincide with a granny's birthday).
(4) Let us assume only one cousin (to the same grandma) to a child being born within a five year period (some will have no cousin at all, some will have several within that period).

Then:

Any particular child has a 1 in 365 chance of being born on their only still living grandma's birthday.

What are the chances that a cousin and grandchild to the same grandma is born within the same 10 minutes?

In five years there are 5 x 365 x 24 x 6 10 minute time slots and that's 262,800.

So the chances that a cousing and grandchild to the same grandma is born within 10 minutes to one particular child is roughly 1 in 250,000.
Now, of course, that has to be multiplied with the chances of the particular child to be born at her only surviving granny's birthday, 1 in 365, leading very roughly to 1 in 100,000,000.

That are the chances that this can happen to one particular child born.

How mayn babies are born each year worldwide? For a rough estimation I have taken the number of babies born in the USA only (4,000,000 per year) have taken 6,000,000,000 as the world's population, have use the knowledge that most places the birthrate is slightly higher than in the USA and came to a rough estimate of 100,000,000 infants born worldwide per year.

With 100,000,000 babies per year born worldwide and each having a 1 in a hundred millions chance to have that particular pattern reported you will very roughly have such an event once per year (none in some years, more than one in others).

The order of magnitude I'm confident is roughly correct. So if you go down to exactly the same minute you'll have it happen about once per ten years. But if you assume that the same coincidence would be equally newsworthy for grandads or for being exactly one year between the two births or... (whatever seems a reportable event) then the probability of one of many different coincidences happening more often then once a year quickly increases.

The probability of the one particular event happening again at least once in the next 1000 years is as close to certain as you could wish.

What goes wrong in many heads is this: The probability of one particular event can be extremely low, so you are correct to feel certain for all practical purposes that it will not happen to you in the next year (e.g., winning the Jackpot of a big national lottery). However, if one looks instead at the probability of a very rare event happening to at least one of many persons at many tries the tiny single event probabilities quickly add up to impressively high probabilities (e.g., the Jackpot being won by any of the Americans going for it).

These high probabilities are notoriously underestimated by most lay persons which can lead to the illusion of seeing above chance patterns even in purely chance events.

Wolfgang