Change = Coins
At least in my area in the US, change is synonymous with coins (of the denominations used in everyday transactions, i.e., for amounts under a dollar; coins of a dollar or more exist but are rarely used and aren't generally referred to as change).
"Change for a dollar" is one dollar worth of coins to exchange for a one dollar note. You could have a big sack of coins, worth significant dollars, and it might be called a bag of change. So in that sense, "change" doesn't necessarily mean a small value, it just means money in the form of coins.
Context for change = small amount
"Change" can be used in a generic way to mean a small amount of something, including something other than money. It's an analogy to how coins are used. In typical cash transactions, paper currency is used for the whole dollar amounts, and coins are used for the fractional-dollar residual amounts. Nobody carries around, or deals with, large quantities of coins. So in everyday usage, people associate "change" with small numbers of coins; "pocket change". That's the usage in the question; a dollar and change is a dollar bill plus some coins, meaning a little more than a dollar.
Something and change
This pattern has been extrapolated to other uses, like minutes and change left in a game, as mentioned by Jason Bassford. Here, "change" similarly refers to the residual amount that is a fraction of the associated unit of measure. But for that context to be recognizable, change must be used in combination with some other explicitly stated unit to which it is compared, like dollars and change or minutes and change. If someone were to say just change left in the game, the meaning wouldn't be understood.
"Change" used in this way doesn't necessarily refer to a small amount of something in absolute terms, but small in relation to something else; a fractional part, which could actually be big in absolute terms if it is a fraction of something big.
For example, Doktor J mentions in a comment -- buying a house for a million and change. Here, the stated unit is million dollars, so change refers to an additional fractional portion of a million dollars.
In typical use, if that fractional portion is a substantial fraction, the amount is expressed in units and fractions; a million and a half, or a minute and a quarter left in the game. The form something and change is typically reserved for when the fraction is small.
However, it is a nebulous usage, so it could also be used to obfuscate. Take a scenario where a customer is inquiring about the price of something. Customer: "What will this cost me?" Salesman: "A hundred and change". Then the salesman explains all of the great benefits and features and convinces the customer to buy it. When the customer goes to pay, the actual amount turns out to be $160.00. The salesman was being purposely vague and stretched the typical meaning without actually lying.
"Change" can be used by itself to indicate a small (relative) amount, and will be understood, if the amount in question is money. For example, I got the house for change. From the context, it is obvious that this is meant figuratively, not literally.
"Change", in this case, would be understood to mean a small amount relative to the value of a house (the implied "unit of measure"); perhaps equivalent to the expression "pennies on the dollar", or exaggeration for emphasis, like saying it was an amount so small it would be transacted with coins.