Yes, this is probably a common experience. I can offer my case as an example, but my general advice is to accept that this may be a long journey and that you may be dealing with potty learning for more than a year. Many cycles of success and regression may be necessary.
Here are some good answers for a similar question.
Our Case
We also started potty training (or "potty learning" as some newer guidelines call it) when my son was around 2 years and 3 months old and did it during a short vacation where we were home for about 5 consecutive days. We followed a popular modern procedure for doing it: low pressure, no bribes, and using diapers only at naptime and bedtime. We would just tell him calmly that pee and poop go in the potty when he did them in his pants.
This seemed to work at first, and we were optimistic about him shortly being fully potty trained in a few weeks. As in your case, he seemed be doing well both at home and at daycare. He even had a couple days at daycare and at home with no accidents. However, this did not last and he began to have 5 or 6 "accidents" some days at home. We kept him out of diapers except when sleeping at home, but the daycare's policy was to put him in a diaper after two "accidents" in a day. I put "accident" in quotes because "accident" seems to imply that urinating or defecating in his pants was fully unintentional. I think that he was often aware that he was peeing and pooping in his pants, but not motivated enough to stop what he was doing and go to the potty. Of course, immature muscle control may have also played a part.
We tried many things, such as 30 minute reminders, allowing him to watch videos that he liked on the potty, and suggesting that he might have "surprise pee" when he said that he didn't need to try the potty after an hour since the last pee. These things all had temporary initial success followed by regression, but there was also a slow overall improvement that kept us working at it.
We tried to never apply too much pressure to avoid negative associations with the potty or accidents. Even if it weren't a bad idea, you cannot force a kid to sit on the potty if they outright refuse.
I feel somewhat confident that he would have been potty trained much sooner if he had not been going to daycare. Several months after beginning, we took a family vacation for over 2 weeks. He improved considerably during this time, and seemed to be accident-free by the end. Unfortunately, he completely regressed again (both at home and at daycare) after returning to daycare.
The Endgame
So we had more than a year of him being sent home with soiled clothes wearing a diaper from daycare almost every day (although the frequency of good days improved) and constantly carrying several wipes and extra pairs of underwear and pants (and also shirts and socks for some leaks). I cleaned up poopy messes in parks and other awkward places without the benefit of a bathtub more times than I can count.
Finally, at about three and a half, we decided to implement a sticker chart where he would earn stars that would get him sugary treats, an episode of his favorite show, or small toys. This seems to be contrary to recommended practice, but we were out of ideas and this seemed to motivate him. However, I'm a scientist, so I have to say that there is no way to know for sure whether this intervention helped or whether there was some other change (muscle control, cognitive development, influence from other kids in his daycare room, etc.) that truly drove the improvement. Probably there were multiple factors.
He started to go several days between accidents, so we started fading out the sticker chart over several weeks (more stickers needed for a reward). He started to forget about putting stars on the chart and seems to have forgotten about what he is supposed to be earning next (for a month or so, the next prize on the sticker chart was constantly on the front of his mind). We traveled last week, and we haven't put the sticker chart back up. He hasn't commented on it yet. So I think that we might have finally arrived. I'm still not convinced we won't have a regression, but I think we maybe out of the woods now.
Takeaways
So what can you learn from this? I'm really not sure. I think we probably started too early. I think he got accustomed to having wet underwear and poop in his underwear before he was truly capable of making it to the potty every time (especially in a daycare environment). Did we suffer unnecessarily for months by not putting him in diapers and insisting on sending him to daycare in underwear, despite getting piles of dirty laundry every day? Maybe we should have just given up and tried again after he turned 3? Maybe we should have started the sticker chart earlier? I don't know.
Some kids became potty trained within in a few weeks at 2 and a half. I know of some kids who still go to daycare in diapers at 4. These outcomes are all probably within the normal variation. Our pediatrician said that we should have a talk if he wasn't potty trained by 3 and a half. You should probably chat with your pediatrician about it at your 3-year check-up.