You are mixing in the issues of measuring time with some sort of unit and of setting the zero of the measurement. You have exactly the same issue for distance. This should not be a problem. If I'm using feet and you meters, then we won't get the same number for a distance. And if I choose one location as the origin, say my house, and you another, say your house, then we won't agree on the numbers involved in coordinates, even if we use the same units of measure.
These should not be particularly difficult. It is no more a problem than trying to understand that getting to school is a different path for you to what it is for your friend who lives on the other side of town. It does not indicate anything strange about space.
This is not what absolute time is referring to.
The thing to look at would be the path from the school to the public library. This path will be the same for both you and your friend, even if you use meters with your house as the origin, and he uses feet with his house as the origin. You may completely disagree about the specific coordinate numbers of the path. This is not a problem for absolute space and absolute time.
Suppose we make observations of some physical process that involves a bunch of events. I will observe those events to happen in some order in time. With absolute space and absolute time we will agree on the ordering. This event happens before that event, and we will agree from our observations. I will observe that the interval between events A and B is longer than the interval between C and D, and in some ratio. You will agree from your observations, including the ratio.
Determining some such ratio is what is most usually meant by measuring time. Suppose we can agree on a standard for what one unit of time is. Then in an absolute space and absolute time, we will agree on the ratio between the size of our standard and the size of some observed interval between events.
Similarly, I will observe the events to be ordered in a direction. Say we pick a direction, say left and right, and we agree on this axis. Then one is to the left of the other one. You will agree from your observations. Event A and B are farther apart than event C and D, and by some ratio. And you will agree from your observations, including the ratio. And if we can agree on a distance unit we will get the same number of that unit for any distance.
Note that part of the idea of absolute space and time is that we can agree on a direction. In some geometries that is a challenge.
In relativity, for example, these are not always true. Sometimes we will disagree about the ordering of events. Sometimes we will disagree about the relative size of the interval between them. For example, if we are moving relative to each other, and if the events are space-like separated, then we may disagree on their ordering in time.