How to approach time data that aren't time series? Imagine a scenario where I managed to, in a three-column data structure, gathered a sample from a single population. The sample consists of a unique identifier for a household, a calendar year, and the number of babies born in the household in that particular calendar year. Data points are only recorded when the number of babies born in the household is $\geq 1$.
We have $N$ data points $\{(u_i, t_i, y_i(u_i, t_i))\}_{i=1}^{N}$, where each $u_i$ is the unique identifier of the $i$th household,  each $t_i$ is a calendar year, and $y_i(u_i, t_i)$ is the number of babies born in the $i$th household in the calendar year $t_i$. The problem of interest is to assess a general trend of total or average births per household in the population over calendar years. Something resembling an effect size and significance test would also be desired as well.
The problem with the above is that it does not neatly fit in the standard time series framework:

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*We don't have an equal number of observations in each calendar year. Not every household will have a recorded observation each year, because not every household will have at least one new baby born each year.

*If my sample is insufficient so that some calendar years were skipped, I will have data for some calendar years but not for others, breaking the usual "equal-spacing" assumption with time series.

*Don't even get me started on trying to figure out whether these data are stationary, no matter how I aggregate them to a single point per calendar year. They probably aren't.

*We could attempt to bin calendar years together, but such a mechanism has no precedent (i.e., assume no similar study has been done), and would be quite arbitrary to implement.

*We also suspect that $\sum_{u_i}y_i(u_i, t_i)$ would be similar (or the sample average) for values of $t_i$ that are close to each other, so any binning of calendar years we do would ignore this to an extent that I would not be comfortable with without adequate justification.

How does one deal with these problems? and perhaps most importantly, are there any literature that can assist with methodology for such a problem?
I suspect that there are tools in Bayesian methods that can help with this, but I want to know where I should focus before I dive in.

Your example look like longitudinal data (also called panel data), that is, many short time series. There is usually not a requirement with equispaced observations for longitudinal data models. On this site, look through panel-data. You could also, in your example, fill in the zeros. Then you have an intermittent time series, see intermittent-time-series.
There is also many posts on unevenly-spaced-time-series. But for your example, maybe look into mixed models for longitudinal data. Forgetting about twin births, you will have Bernoulli data, so maybe a binomial mixed model. Or another possibility: Look at the interevent times!
