Do you see the stamping on the data plate that says 'Capacity: 3/8" 10mm'? Do you see the speed range stamped on the same plate?
Both are there for a reason: they are the parameters within which the manufacturer recommends the machine be used.
If you are to use it at 60% over that capacity (16mm), do you think it will live long and prosper?
If you are to use it to drill at 16mm diameter, what is a sensible speed for that operation in different materials?
https://www.sserc.org.uk/wp-content/uploads/2011/12/drill-speed-chart.pdf
Do you have an existing chuck for it? That will likely tell you the taper required. Common drill spindle tapers shown here:
https://www.arceurotrade.co.uk/projects/Info/Drill-Chuck-Taper-Dimensions.html
It looks like your caliper is saying 10.25mm, so B12 is close, but you need to confirm. Searching for B12 drill chuck does not easily find anything over 10mm capacity*. Hence, we return to the reason the data plate is stamped as it is.
* There is this one:
https://www.aptoolsltd.co.uk/shop/r...6654-6655-6656-6657-6658-6659-6660-6661-6662/
Edit: thinking some more, the imperial scale on the caliper looks closer to 1/2" so it might be JT2.