In my project we assign 4 hours for 1 story point and estimate based on story point. So in this case one ideal day of work will be done in 2 elapsed days. From an agile standpoint are we doing story point estimation or ideal days estimation or none of these?
You're definitely doing a time based estimation, hence, no story points in the original sense. Story points are meant to be an abstract measure for relative estimation of story complexity. The mapping to an amount of time only happens implicitly, when you pull a certain number of story points into your fixed-time iteration. Here's an interesting discussion about how Story Points relate to hours.
But, on the other hand, your not really doing ideal-days estimation either, because at least you say that you estimate in hours. Ideal days means that you give estimates like "in 1.5 days we will be through with this". Of course you can map 8h = 2SP = 1 day. Actually, I would bet that many of you estimate in days and then derive hours from that and than story points. So implicitly you're probably doing ideal-days estimation ;)
I too agree with the answer above. Most of the people who follow agile misinterpret estimates as 'hours required to complete the task". This could be because most of these people are moving from traditional waterfall to agile and they are very much into keeping track of all the hours spent on a particular chunk of work. Instead of estimating just relatively size all the stories wrt each other. When we look at a group of stories it is pretty easy to compare them relatively to each other and have a good understanding of which ones are similar. The hard part is to predict how fast we will finish each story.
Now you'll ask - Why does it matter if we size our story if we are not mentioning the hours of work?
Well when you talk about estimates to client, they naturally start to think about time and cost. This leads to change in number of points because 'it is taking longer than estimate'. Just because a story has taken longer than anticipated, does that mean its relative size is different to all the other stories? Maybe… but most often it is that our velocity is not what we initially thought it would be. Thus when we talk about story size, we are trying to focus on velocity being the value different from expectation and not the story points. This way we'll move from updating story points from time to time and will try to improve the velocity.