Is 337 the same as SR416SW?

Yes. 337 is another code for the same cell: silver oxide, 1.55 V, 4.8 × 1.6 mm.

Can I use LR416 instead of SR416SW?

Works, but runs out sooner. Avoid in watches and medical devices.

SR416SW is a 1.55 V silver oxide cell with a typical capacity of 8 mAh.

4.8 mm across and 1.6 mm thick.

SR416SW is the same cell as 337, so any of those codes will fit; LR416 is the same size in alkaline and will work, though it runs out sooner.

IEC designation SR416.

SR416SW at actual relative size: 4.8 mm across, 1.6 mm thick. The dashed circle is a US quarter, 24.26 mm across (US Mint coin specifications).

The same cell under another code. Any of these fits, whatever the brand.

Same size, different chemistry. Read the verdict before you buy.

Cell	Chemistry	Voltage	Verdict
LR416	alkaline	1.5 V	Downgrade Works, but runs out sooner. Avoid in watches and medical devices.

On the shelf it is sold with a SR416 on the pack .

Is 337 the same as SR416SW?: Yes. 337 is another code for the same cell: silver oxide, 1.55 V, 4.8 × 1.6 mm.
Can I use LR416 instead of SR416SW?: Works, but runs out sooner. Avoid in watches and medical devices.
What voltage is SR416SW?: SR416SW is a 1.55 V silver oxide cell with a typical capacity of 8 mAh.
What size is SR416SW?: 4.8 mm across and 1.6 mm thick.