Is there any sort of in-universe scientific explanation of the irregular seasons in "A Song of Ice and Fire". What does the orbit/tilt of their planet look like? How can a year be measured?
As I read in an interview with the author, he said the explanation was magical and would be further explained in the last book.
I have a theory for the second part of your question. The planet of "A Song of Ice and Fire" has a moon. Character's often speak of the "moon's blood" as a clear euphemism for menstruation, which is a reasonable clue that a lunar cycle is at least as close to a month as it is for Luna and Earth. A year could be measured in terms of lunar cycles.
The seafaring sections of the book also make several mention of the tides, which would imply that the moon makes a fairly rapid orbit of ASOIAF's planet. Luna orbiting Earth ~once per day is what causes a 12h tidal cycle.
Having a lunar calendar makes it a lot easier for a common person with no training in astronomy to observe and measure the year. As you mention, it does seem that nearly everyone keeps track of their age & name-day. Even the wildlings have a tradition of not naming their children until they're 2 years old. (Not that the wildlings are dull or ignorant, but it would be hard to argue that another culture were less 'educated' than the wildlings.) After all, it's a lot easier to count to 12 than to 365...
There were possibly regular seasons in ancient history, even further back than the Age of Heroes.
Writing and records were rare even back then and most of what we know from that time is legends passed on by story. The Maesters have looked into the matter but have not been able to conclude anything as there is just not enough information from that long ago...
Though the Citadel has long sought to learn the manner by which it may predict the length and change of seasons, all efforts have been confounded. Septon Barth appeared to argue, in a fragmentary treatise, that the inconstancy of the seasons was a matter of magical art rather than trustworthy knowledge. Maester Nicol's The Measure of the Days—otherwise a laudable work containing much of use—seems influenced by this argument. Based upon his work on the movement of stars in the firmament, Nicol argues unconvincingly that the seasons might once have been of a regular length, determined solely by the way in which the globe faces the sun in its heavenly course. The notion behind it seems true enough—that the lengthening and shortening of days, if more regular, would have led to more regular seasons—but he could find no evidence that such was ever the case, beyond the most ancient of tales.
The World of Ice and Fire - Ancient History: The Long Night
The "in-universe" reasoning seems to come from GRRM's quote on the matter:
I asked "Is magic coming back into the world because there are dragons, or are dragons coming back into the world because there's magic?" George said, "Yes. Hmm, there's excellent cheese on that pizza!" (in combination with some of the stuff he said on a panel this morning, I take it to mean that the seasons, winter and summer, are magical in nature, and he's going to reveal what it's all about eventually, but not yet.)
Which was further confirmed by the co-authors of The World of Ice and Fire
But what causes these strange, unpredictable seasons? We know for certain that they’re not in any way predictable, at least not with the knowledge and observations of the people in the setting. These are “proper” seasons, though, that much we do know. If it’s summer in Westeros, it’s summer in the rest of the hemisphere, too. And yes, it really does seem to have something to do with axial tilt, much as our seasons do. It’s noted that winter means that the days grow shorter. It’s not simply that the weather becomes really cold or really warm, the planet itself appears to change its orbital dynamics in very strange and unpredictable ways.
It’s been a popular topic on the A Song of Ice and Fire forums, this whole matter of what causes the weird seasons. Suggested theories have ranged as far as suggesting dark planets in the near vicinity, perhaps a binary star, and more. But it’s rather fruitless; the author is prosaic on the topic and has provided the direct answer: it’s magic, trying to figure out a scientific, realistic explanation is bound to fail. If the magic means that some sorcerous force works on a planet-wide scale to tilt the planet this way or that… well, that’s what it means. Or is it? Can there be some combination of physical causes that would approximate the apparent-unpredictability and lengthiness of the seasons? I’ve yet to see someone manage anything convincing, but it may be an interesting puzzle for the more scientifically inclined.
I am actually not sure that the long summers and winters are in effect outside of Westeros. I don't think that the same seasons are in effect in the Nine Free Cities and in Vaes Dothrak. Therefore, the long seasons are not on the whole planet, and I think it rules out an astrophysical cause.
It looks like it's really caused by magic, and I would assume it has something to with the White Walkers.
It's been a while since I read the books, but I thought they had regular yearly seasonal cycles, but that these cycles were overlaid on the longer-term seasons. So, in a Winter season, you'd get warmer and colder days, but overall, it'd be much colder than during a Summer season. So, they'd be more like short-term ice ages & warming periods, than actual seasons.
As to what could cause that? Periodic variation in volcanic activity, maybe?
"Winter is coming"
Veselin Kostov, Daniel Allan, Nikolaus Hartman, Scott Guzewich, Justin Rogers
Those that do not sow care little about such mundane things as equinoxes or planting seasons, or even crop rotation for that matter. Wherever and whenever the reavers reave, the mood is always foul and the nights are never warm or pleasant. For the rest of the good folks of Westeros, however, a decent grasp of the long-term weather forecast is a necessity. Many a maester have tried to play the Game of Weather Patterns and foretell when to plant those last turnip seeds, hoping for a few more years of balmy respite. Tried and failed. For other than the somewhat vague (if not outright meaningless) omens of "Winter is Coming", their meteorological efforts have been worse than useless. To right that appalling wrong, here we attempt to explain the apparently erratic seasonal changes in the world of G.R.R.M. A natural explanation for such phenomena is the unique behavior of a circumbinary planet. Thus, by speculating that the planet under scrutiny is orbiting a pair of stars, we utilize the power of numerical three-body dynamics to predict that, unfortunately, it is not possible to predict either the length, or the severity of any coming winter. We conclude that, alas, the Maesters were right -- one can only throw their hands in the air in frustration and, defeated by non-analytic solutions, mumble "Coming winter? May be long and nasty (~850 days, T<268K) or may be short and sweet (~600 days, T~273K). Who knows..."