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A caveat

This is definitely not a conclusive answer, and there are other possible interpretations along similar lines: "shilling", or just the shilling symbol, might have colloquially been used for 12 cents (in the same way "penny" colloquially means 1 cent), or a dime (a division between cents and dollars in the same way a shilling was between pence and pounds), or some other obsolete coin in local use.

As found by Pieter Geerkens, the instructions for the 1860 census clearly state that values should be written in dollars only. However, from context, we can quickly rule out $12 and $10 as plausible values for these columns, so the logical conclusion is that the clerk failed to obey that instruction for these values, and wrote the prices in terms that he was familiar with in every day life.

These weren't going to disappear overnight, so we can assume that in 1860 many people were still accustomed to them in every day life.

As found by Pieter Geerkens, the instructions for the 1860 census repeatedly state that values should be written in dollars only. However, from context, we can quickly rule out $12 and $10 as plausible values for these columns, so the logical conclusion is that the clerk failed to obey that instruction for these values, and wrote the prices in terms that he was familiar with in every day life.

These weren't going to disappear overnight, so we can assume that in 1860 many people were still accustomed to them in every day life. This would also make sense of the reminders in the census instructions to write values in dollars - today, it would mostly go without saying, since the US Dollar is firmly established as the local currency.

The symbol

The Spanish "shilling"

The currencies specifically mentioned in the 1857 Act are not British currency, but the Spanish and Mexican dollar. These were the basis of the US dollar, and were widely used at the time. Two things can be seen in the Act:

The value

Not following instructions

As found by Pieter Geerkens, the instructions for the 1860 census clearly state that values should be written in dollars only. However, from context, we can quickly rule out $12 and $10 as plausible values for these columns, so the logical conclusion is that the clerk failed to obey that instruction for these values, and wrote the prices in terms that he was familiar with in every day life.

The symbol

American shillings

The Merriam-Webster entry for "shilling" includes this broad definition:

any of several early American coins

This confirms that the term was in colloquial use, and even in some cases the official name of coins with varying values.

The Spanish "shilling"

The currencies specifically mentioned in the 1857 Act are not British currency, but the Spanish and Mexican dollar. These were the basis of the US dollar, and were widely used at the time. Two noteworthy things can be seen in the Act:

The possible value