Is this properly composted horse manure? Shouldn't be any live seeds in that.

Looks like an oxalis - the kind that spreads by runners and has exploding seed pods. I have it here - the seeds are tiny and also stick to things.

Might it have hitchhiked in on the strawberries?

Don't see that as clover. It looks like an annul that I have. Mine bloom yellow and are vertical. Perhaps yours are small young plants that are short yet. Pull them up getting the root.

Some pics - thinking it's likely oxalis stricta...

https://hort.extension.wisc.edu/articles/common-yellow-woodsorrel-oxalis-stricta/

Yes. It is Oxalis - perennial and persistent. Spreads by bulbils as well as seeds. Extremely hard to eradicate. And not a clover.

Of course, that identification makes your horse manure innocent. Oxalis is poisonous for horses. Moreso even, I think, than for people. Clover, on the other hand, is good feed stock for horses. One might wonder if the shredded straw got contaminated with the stuff.

IMPE, this particular oxalis doesn't form bulbils, but spreads by underground runners and shooting seeds. The link I shared also mentions that.

The one I have here is not easy to pull - it's quite tenacious and you often need to use a tool to dig out the runners, since they like to snap off and keep growing and spreading.

Oxalis species are not, per se, poisonous for people, unless one eats mountains of them. O stricta, for example, is a tasty citrus flavoured addition to a salad, and one species, Oxalis tuberosa, is a major vegetable in some parts of the world.

O stricta does produce some bulbils in my garden. But, as you say, it does shoot seeds and runs.

Agreed, Oxalis. The seeds probably hitchhiked on the shredded straw.

Unfortunately, you will now have a hard time eradicating it. I don't see any of their okra-like seed pods (yet) so pull the plants out now before they form. Getting all of the roots out will take some persistence, I battle these in a couple of my beds.